Thermomechanical Response of Shape Memory Alloy Hybrid ...me to the topic of shape memory alloys and...

Thermomechanical Response ofShape Memory Alloy Hybrid Composites

by

Travis L. Turner

Dissertation Submitted to the Faculty of theVirginia Polytechnic Institute and State University

in Partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophyin

Engineering Mechanics

APPROVED:

Mahendra P. Singh, Chairman Christopher R. Fuller

Robert A. Heller Robert M. Jones

Saad A. Ragab Stephen A. Rizzi

November 17, 2000Blacksburg, Virginia

Keywords: Nitinol, embedded actuators, hybrid compositefabrication, nonlinear thermoelasticity, geometric nonlinearity

Abstract

Thermomechanical Response of

Shape Memory Alloy Hybrid Composites

Travis L. Turner

Committee Chairman: Mahendra P. Singh

Engineering Mechanics

This study examines the use of embedded shape memory alloy (SMA) actuators for adaptive

control of the themomechanical response of composite structures. Control of static and dynamic

responses are demonstrated including thermal buckling, thermal post-buckling, vibration, sonic

fatigue, and acoustic transmission. A thermomechanical model is presented for analyzing such

shape memory alloy hybrid composite (SMAHC) structures exposed to thermal and mechanical

loads. Also presented are (1) fabrication procedures for SMAHC specimens, (2) characterization

of the constituent materials for model quantification, (3) development of the test apparatus for

conducting static and dynamic experiments on specimens with and without SMA, (4) discussion

of the experimental results, and (5) validation of the analytical and numerical tools developed

in the study.

The constitutive model developed to describe the mechanics of a SMAHC lamina captures

the material nonlinearity with temperature of the SMA and matrix material if necessary. It is in

a form that is amenable to commercial finite element (FE) code implementation. The model is

valid for constrained, restrained, or free recovery configurations with appropriate measurements

of fundamental engineering properties. This constitutive model is used along with classical

lamination theory and the FE method to formulate the equations of motion for panel-type

structures subjected to steady-state thermal and dynamic mechanical loads. Mechanical loads that

are considered include acoustic pressure, inertial (base acceleration), and concentrated forces.

Four solution types are developed from the governing equations including thermal buckling,

thermal post-buckling, dynamic response, and acoustic transmission/radiation. These solution

procedures are compared with closed-form and/or other known solutions to benchmark the

numerical tools developed in this study.

Practical solutions for overcoming fabrication issues and obtaining repeatable specimens are

demonstrated. Results from characterization of the SMA constituent are highlighted with regard

to their impact on thermomechanical modeling. Results from static and dynamic tests on a

SMAHC beam specimen are presented, which demonstrate the enormous control authority of the

SMA actuators. Excellent agreement is achieved between the predicted and measured responses

including thermal buckling, thermal post-buckling, and dynamic response due to inertial loading.

The validated model and thermomechanical analysis tools are used to demonstrate a variety of

static and dynamic response behaviors associated with SMAHC structures. Topics of discussion

include the fundamental mechanics of SMAHC structures, control of static (thermal buckling and

post-buckling) and dynamic responses (vibration, sonic fatigue, and acoustic transmission), and

SMAHC design considerations for these applications. The dynamic response performance of a

SMAHC panel specimen is compared to conventional response abatement approaches. SMAHCs

are shown to have significant advantages for vibration, sonic fatigue, and noise control.

Dedication

This work is dedicated to my wife, Andrea Turner, whose love and support has exceeded all

imagination and who has made our life together very rich; to my father and mother, Lee, Jr. and

Mary-John Turner, who instilled in me the qualities to strive for and achieve this goal and who

have given so much in every respect to ensure their children’s happiness and success; and to my

father and mother-in-law, Richard and Joan Peters, whose boundless support and encouragement

I will always greatly appreciate. I am a very lucky man to have such a wonderful family. I

pray that completion of this work somehow justifies the sacrifices that have been made by all.

Although this journey has been a good one, I hope the best for all of us is yet to come.

iv

Acknowledgements

I would like to take this opportunity to thank all who have contributed to the completion of

this work. First, I would like to thank my advisor, Dr. Mahendra Singh, for his technical advice

and support. He helped make my academic experience at VPI&SU very rewarding and he has

been a pleasure to work with throughout this endeavor. I would also like to thank the other

committee members; Dr. Christopher Fuller, Dr. Robert Heller, Dr. Robert Jones, Dr. Saad

Ragab, and Dr. Stephen Rizzi for their helpful insights and suggestions. In particular, I would

like to recognize Dr. Stephen Rizzi for his support and countless technical interchanges over the

years. I would also like to thank Dr. Chuh Mei, of Old Dominion University, for introducing

me to the topic of shape memory alloys and for many helpful discussions.

I am grateful to NASA LaRC for the opportunity to pursue graduate studies while maintaining

employment. I thank my supervisors, Dr. Kevin Shepherd and Dr. Richard Silcox, for their

support and for allowing me the time and latitude to complete this work.

I gratefully acknowledge the assistance of Cynthia Lach, R. Keith Bird, Craig Ohlhorst

(NASA LaRC), William Johnston, Joel Alexa, Stewart Walker (Analytical Services and Mate-

rials), Jerry Draper (Lockheed Martin Astronautics), and Kenneth Blount (Materials Research

and Engineering) for their help in characterizing the thermomechanical properties of the glass-

epoxy and Nitinol materials. Thanks are due to Scott Wallace (NASA LaRC) and Dr. Ravi

Shenoy (Analytical Services and Materials) for their assistance in providing differential scanning

calorimetry measurements from the Nitinol. The assistance of Roberto Cano and Hoa Luong

(NASA LaRC) in lending their composite fabrication expertise is also very much appreciated. I

thank Howard Jones and Jeff Pierro for their help in dynamic testing of the SMAHC specimens.

Finally, I would like to thank my family and friends for their endless patience, love, and

support. I feel truly blessed to be able to share life’s experiences with such great people.

v

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Dedication. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Acknowledgements .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

List of Figures . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xviii

Chapter 1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Background and Motivation. . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Scope and Objectives . . .. . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Research Organization . .. . . . . . . . . . . . . . . . . . . . . . . . . . 6

Chapter 2 Background on SMAs and Constitutive Modeling . .. . . . . . . . . . . 9

2.1 Background on Shape Memory Alloys. . . . . . . . . . . . . . . . . . . 9

2.2 Constitutive Modeling . . .. . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Deformation Behavior and Stress Resultants. . . . . . . . . . . . . . . 26

Chapter 3 Specimen Fabrication and Material System Characterization. . . . . . 30

3.1 Material System Selection. . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Specimen Fabrication . . .. . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3 Material System Characterization . . .. . . . . . . . . . . . . . . . . . 40

3.3.1 Glass-Epoxy Matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3.2 Nitinol Alloy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

vi

Chapter 4 Finite Element Formulation. . . . . . . . . . . . . . . . . . . . . . . . . 70

4.1 Element Displacement Functions . . .. . . . . . . . . . . . . . . . . . 70

4.2 Euler-Lagrange Equations. . . . . . . . . . . . . . . . . . . . . . . . . 74

4.2.1 Separation of Static and Dynamic Components . . .. . . . . . . . . . 85

Chapter 5 Static Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.1 Static Stability and Nonlinear Analysis. . . . . . . . . . . . . . . . . . 96

5.2 Thermal Buckling Analysis. . . . . . . . . . . . . . . . . . . . . . . . 100

5.2.1 Constant Matrix Material Properties . .. . . . . . . . . . . . . . . . . 101

5.2.2 Non-Constant Matrix Material Properties. . . . . . . . . . . . . . . . 104

5.3 Thermal Post-buckling Analysis. . . . . . . . . . . . . . . . . . . . . 107

5.4 Analytical Validation . . .. . . . . . . . . . . . . . . . . . . . . . . . 113

5.4.1 Simply-Supported Isotropic Beam . . .. . . . . . . . . . . . . . . . . 113

5.4.2 Simply-Supported and Clamped Isotropic Plates . . .. . . . . . . . . 115

5.4.3 Simply-Supported Composite Plate . .. . . . . . . . . . . . . . . . . 117

5.5 Experimental Validation . .. . . . . . . . . . . . . . . . . . . . . . . . 119

5.5.1 Clamped SMAHC Beam .. . . . . . . . . . . . . . . . . . . . . . . . 119

Chapter 6 Dynamic Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.1 Equations of Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.1.1 Acoustic Excitation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

6.1.2 Concentrated Force Excitation. . . . . . . . . . . . . . . . . . . . . . 128

6.1.3 Inertial Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

vii

6.2 Eigen-solution and Modal Coordinates. . . . . . . . . . . . . . . . . 130

6.3 Continuous Spectral Analysis — Random Response. . . . . . . . . 131

6.4 Discrete Spectral Analysis. . . . . . . . . . . . . . . . . . . . . . . . 134


6.5.1 Simply-Supported Aluminum Beam . .. . . . . . . . . . . . . . . . . 140

6.5.2 Simply-Supported Aluminum Panel . .. . . . . . . . . . . . . . . . . 143


6.6.1 Clamped Aluminum Beam. . . . . . . . . . . . . . . . . . . . . . . . 153

6.6.2 Clamped Glass-Epoxy Beam. . . . . . . . . . . . . . . . . . . . . . . 158

6.6.3 Clamped SMAHC Beam .. . . . . . . . . . . . . . . . . . . . . . . . 159

6.6.4 Clamped Aluminum Panel. . . . . . . . . . . . . . . . . . . . . . . . 173

Chapter 7 Acoustic Radiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

7.1 Continuous Spectral Analysis. . . . . . . . . . . . . . . . . . . . . . 185

7.2 Discrete Spectral Analysis. . . . . . . . . . . . . . . . . . . . . . . . 189


7.3.1 Simply-Supported Aluminum Panel . .. . . . . . . . . . . . . . . . . 194


7.4.1 Clamped Aluminum Panel. . . . . . . . . . . . . . . . . . . . . . . . 202

Chapter 8 Applications and Demonstrations . . .. . . . . . . . . . . . . . . . . 206

8.1 General Observations . . .. . . . . . . . . . . . . . . . . . . . . . . . 207

8.2 Thermal Buckling and Post-buckling .. . . . . . . . . . . . . . . . . 210

8.3 Dynamic Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

8.4 Sonic Fatigue and Acoustic Transmission. . . . . . . . . . . . . . . 221

viii

Chapter 9 Summary and Concluding Remarks . .. . . . . . . . . . . . . . . . . 228

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

Appendix A RSM Hybrid Lamina Constitutive Relations. . . . . . . . . . . . . . 241

Appendix B Gauss-Legendre Numerical Integration. . . . . . . . . . . . . . . . . 245

B.1 Element Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

B.2 Mean Square Radiated Acoustic Pressure. . . . . . . . . . . . . . . . 247

B.3 Radiated Acoustic Power .. . . . . . . . . . . . . . . . . . . . . . . . 248

Appendix C Closed Form Evaluation of Element Area Integrals .. . . . . . . . . 250

Appendix D Radiation Far-Field Approximations . .. . . . . . . . . . . . . . . . . 256

Appendix E Acoustic Intensity, Power, and TL . .. . . . . . . . . . . . . . . . . 258

Appendix F Continuous Spectral Examples. . . . . . . . . . . . . . . . . . . . . . 260

F.1 Harmonic Excitation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

F.2 Band-Unlimited White Noise Excitation. . . . . . . . . . . . . . . . 262

Appendix G Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

ix

List of Figures

Figure 2.1 Schematic of martensitic and reverse transformations, after Liang and

Rogers [13]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Figure 2.2 Schematic of the shape memory effect, after Wayman and Duerig in

[32]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Figure 2.3 Nitinol recovery stress versus temperature and initial strain, after Cross

et al. [28]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Figure 2.4 Nitinol Young’s modulus versus temperature, after Cross et al. [28]. . 13

Figure 2.5 Stress-strain curves for a shape memory alloy at various temperatures,

after Funakubo [33]. . . .. . . . . . . . . . . . . . . . . . . . . . . . . 15

Figure 2.6 Stress-strain curves for a shape memory alloy at various temperatures,

after Wayman and Duerig in [32].. . . . . . . . . . . . . . . . . . . . . 16

Figure 2.7 Variation in austenite start temperature with weight % Ni in Nitinol,

after Buehler and Wang [1].. . . . . . . . . . . . . . . . . . . . . . . . 17

Figure 2.8 Representative volume element for a SMAHC lamina.. . . . . . . . . 19

Figure 3.1 Schematic of beam specimens cut from a SMA hybrid composite

laminate. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Figure 3.2 Packaging strain recovery rig.. . . . . . . . . . . . . . . . . . . . . . . 34

Figure 3.3 Nitinol ribbon gripping apparatus for prestraining and mechanical

testing. . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Figure 3.4 Nitinol ribbon prestrain and mechanical testing apparatus.. . . . . . . 36

Figure 3.5 Mechanical drawing of the laminate lay-up and cure tooling apparatus

(all dimensions in inches).. . . . . . . . . . . . . . . . . . . . . . . . . 37

Figure 3.6 Laminate lay-up procedure at the stage of completing layer 2. . . . . 38

x

Figure 3.7 Completed laminate with cure control thermocouples and NiTi ribbon

grips torqued. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Figure 3.8 Laminate after vacuum-bagging and applying vacuum.. . . . . . . . . 39

Figure 3.9 Consolidated SMAHC laminated panel.. . . . . . . . . . . . . . . . . . 39

Figure 3.10 SMAHC beam specimens machined from panel. . .. . . . . . . . . . 40

Figure 3.11 Schematic of the CTE specimen geometry for measurement by the

interferometric dilatometer method. . . .. . . . . . . . . . . . . . . . . 43

Figure 3.12 Fiberite E-glass/934 epoxy longitudinal (fiber-direction) CTE vs.

temperature by two methods.. . . . . . . . . . . . . . . . . . . . . . . 44

Figure 3.13 Fiberite E-glass/934 epoxy transverse CTE vs. temperature by two

methods. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Figure 3.14 Schematic of the machining process for removing tensile, CTE, and

SMAHC beam specimens from the SMAHC laminate.. . . . . . . . . 47

Figure 3.15 Heat-first DSC signatures for first- and second-heat of as-received

material. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Figure 3.16 Cool-first DSC signatures for first- and second-cool of as-received

material. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Figure 3.17 Max recovery stress versus thermal cycle number for Nitinol ribbon

samples with 2%, 3%, and 4% prestrain.. . . . . . . . . . . . . . . . . 54

Figure 3.18 Recovery stress versus temperature for Nitinol ribbon samples with

2% and 4% prestrain, after 50 cycles. .. . . . . . . . . . . . . . . . . 55

Figure 3.19 Recovery stress versus thermal cycle and temperature for sample with

2% prestrain.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Figure 3.20 Recovery stress versus thermal cycle temperature for sample with 4%

prestrain. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

xi

Figure 3.21 Applied stress versus strain for 0.09x0.006x10 inch sample with 0%

prestrain (as-received) at ambient temperature and 120�C. . . . . . . . 59

Figure 3.22 Applied stress versus strain for 0.09x0.006x10 inch sample with 4%

prestrain at ambient temperature and 120�C. . . . . . . . . . . . . . . . 60

Figure 3.23 Apparatus for installation of a Nitinol ribbon sample in a tensile test

machine for recovery force measurement.. . . . . . . . . . . . . . . . 62

Figure 3.24 Recovery force measurement in progress.. . . . . . . . . . . . . . . . 63

Figure 3.25 Recovery stress versus temperature of the Nitinol ribbon (prestrained

4% and released) for thermal cycles 2–4 and 50. . .. . . . . . . . . . 64

Figure 3.26 Recovery stress versus temperature of the Nitinol ribbon for 3% prestrain

constructed from the LMA data for thermal cycles 2–4 and 50. . . . 65

Figure 3.27 Modulus for the Nitinol ribbon prestrained 4% and released as a

function of temperature. .. . . . . . . . . . . . . . . . . . . . . . . . . 67

Figure 4.1 Schematic of 24 dof rectangular plate element. . . .. . . . . . . . . . 71

Figure 4.2 Overall thermomechanical analysis flowchart.. . . . . . . . . . . . . . 95

Figure 5.1 Thermal buckling calculation flowchart.. . . . . . . . . . . . . . . . 106

Figure 5.2 Thermal post-buckling calculation flowchart.. . . . . . . . . . . . . 112

Figure 5.3 Normalized maximum deflection versus dimensionless thermal load for

a clamped, 0.102x0.102x0.001 m (4x4x0.040 inch), aluminum plate

with uniform thermal load.. . . . . . . . . . . . . . . . . . . . . . . . 116

Figure 5.4 Normalized maximum deflection versus normalized temperature change

for a simply-supported, 0.152x0.152x0.001 m (6x6x0.040 inch),

(±45/0/90)s graphite-epoxy laminate with uniform thermal load. . . 118

Figure 5.5 Front view of the SMAHC beam specimen mounted in clamping

fixture used for static and dynamic testing.. . . . . . . . . . . . . . 120

xii

Figure 5.6 Experimental configuration for measurement of the static thermoelastic

response of a SMAHC beam specimen.. . . . . . . . . . . . . . . . 121

Figure 5.7 Measured and predicted thermal post-buckling deflection (normalized

to the beam thickness) at the mid-span of the SMAHC beam. . . . 122

Figure 6.1 Schematic of acoustically excited panel.. . . . . . . . . . . . . . . . 127

Figure 6.2 Schematic of simply-supported beam subject to planar acoustic,

concentrated force, and base acceleration excitation.. . . . . . . . . 140

Figure 6.3 Schematic of simply-supported panel subjected to planar acoustic,

concentrated force, and base acceleration excitation.. . . . . . . . . 143

Figure 6.4 Classical and finite element displacement response PSDs at x=0.102, y=

0.076 m (x=4, y=3 inches) versus frequency for inertial loading case. 146

Figure 6.5 Classical and finite element displacement response PSDs at x=0.102,

y= 0.076 m (x=4, y=3 inches) versus frequency for normal acoustic

pressure loading case. . . .. . . . . . . . . . . . . . . . . . . . . . . . 147


y= 0.076 m (x=4, y=3 inches) versus frequency for concentrated force

loading case.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148


y= 0.076 m (x=4, y=3 inches) versus frequency for�i=60�, �i=45�

acoustic pressure loading case.. . . . . . . . . . . . . . . . . . . . . . 149

Figure 6.8 Overall configuration for base acceleration tests on beam specimens. 152

Figure 6.9 Aluminum beam specimen mounted in base acceleration test

configuration.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

Figure 6.10 Measured and predicted displacement PSD comparison for position 1

on the aluminum beam (�f=0.25 Hz). . . . . . . . . . . . . . . . . . 156

xiii


on the aluminum beam (�f=0.25 Hz). . . . . . . . . . . . . . . . . . 157


on the glass-epoxy beam (�f=0.25 Hz). . . . . . . . . . . . . . . . . 160


on the glass-epoxy beam (�f=0.25 Hz). . . . . . . . . . . . . . . . . 161

Figure 6.14 Front view of the SMAHC beam specimen mounted in clamping

fixture for dynamic testing.. . . . . . . . . . . . . . . . . . . . . . . . 162

Figure 6.15 Rear view of the SMAHC beam specimen mounted in clamping fixture

for dynamic testing.. . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Figure 6.16 Measured mode 1 and mode 3 natural frequencies of the SMAHC

beam versus temperature. .. . . . . . . . . . . . . . . . . . . . . . . . 165

Figure 6.17 Measured displacement PSDs for position 2 on the SMAHC beam at

ambient and elevated temperatures (�f=0.25 Hz). . . . . . . . . . . 167


on the SMAHC beam at ambient temperature (�f=0.25 Hz). . . . . 169


on the SMAHC beam at 71.1�C (160�F, �f=0.25 Hz). . . . . . . . 170





Figure 6.22 Measured and predicted displacement PSD comparison for the SMAHC

beam at ambient temperature with boundary effects (�f=0.25 Hz). . 174

xiv


beam at 71.1�C with boundary effects (160�F,�f=0.25 Hz). . . . . 175


beam at 93.3�C with boundary effects (200�F,�f=0.25 Hz). . . . . 176


beam at 121.1�C with boundary effects (250�F,�f=0.25 Hz). . . . 177

Figure 6.26 Transmission loss apparatus schematic.. . . . . . . . . . . . . . . . . 178

Figure 6.27 PSD of the acoustic excitation incident upon the panel (OASPL 109

dB, ref 20�Pa). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

Figure 6.28 Comparison of predicted and measured panel normal velocity PSD at

coordinates of x=0.152, y=0.102 m (x=6, y=4 inches) relative to panel

lower left corner. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

Figure 6.29 Comparison of predicted and measured panel normal velocity PSD at

coordinates of x=0.229, y=0.051 m (x=9, y=2 inches) relative to panel

lower left corner. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

Figure 7.1 Schematic of a panel radiating to a semi-infinite acoustic field. . . . 185

Figure 7.2 Normal versus tangential radiated acoustic intensity R=3.05 m (120

inches),�=[0,2�], �=0 for normal acoustic incidence, 10–1000 Hz. 196


inches),�=[0,2�], �=0 for normal acoustic incidence, 800 Hz. . . . 197

Figure 7.4 Transmission loss versus frequency for normal acoustic incidence,

10–1000 Hz.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198


inches),�=[0,2�], �=0 for �i=60, �i=45 acoustic incidence, 10–1000

Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

xv


inches),�=[0,2�], �=0 for �i=60,�i=45 acoustic incidence, 800 Hz. 200

Figure 7.7 Transmission loss versus frequency for�i=60, �i=45 acoustic

incidence, 10–1000 Hz. . .. . . . . . . . . . . . . . . . . . . . . . . . 201

Figure 7.8 Comparison of the predicted and measured transmitted pressure PSD

at the center microphone of array location R1 (see Table 7.2 and

Figure 6.26).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

Figure 7.9 Comparison of the predicted and measured transmitted pressure PSD

at bottom microphone of array location R3 (see Table 7.2 and Figure

6.26). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

Figure 8.1 Non-deformed/reference (gray-shade) and deformed (mesh) in-plane

geometries for 0.457x0.025x0.001 m (18x1x0.040 in.) cantilevered

and clamped-clamped glass-epoxy and SMAHC beams. Displacements

calculated at 65.56�C (150�F) and magnified 1000 times.. . . . . 209

Figure 8.2 Normalized mid-span post-buckling deflection versus temperature

change for conventional and SMAHC 0.457x0.025x0.001 m

(18x1x0.040 in.) clamped beams.. . . . . . . . . . . . . . . . . . . . 212


change for SMAHC 0.457x0.025x0.001 m (18x1x0.040 in.) clamped

beams with SMA in all/0� layers. . . . . . . . . . . . . . . . . . . . . 214


change for composite 0.457x0.025x0.001 m (18x1x0.040 in.) clamped

beams with various SMA volume fractions.. . . . . . . . . . . . . . 216

Figure 8.5 Post-buckling deflection of SMAHC laminates versus�T andva. . 217

Figure 8.6 Wmax PSD for the baseline and SMAHC laminates.. . . . . . . . . 219

xvi

Figure 8.7 Wmax PSD for a SMAHC laminate at three temperatures.. . . . . . 220

Figure 8.8 Wmax PSD comparison of two SMAHCs with conventional

treatments. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

Figure 8.9 TL versus frequency for the baseline and SMAHC laminates. . . . . 226

Figure 8.10 TL comparison of two SMAHCs with conventional treatments. . . . 227

Figure A.1 Representative volume element for SMA fiber-reinforced hybrid

composite lamina.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

Figure B.1 Schematic of plate element in centroidal non-dimensional

coordinates. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

Figure F.1 Schematic of upper-half-plane contour for evaluation of dynamic

response due to band-unlimited white noise excitation.. . . . . . . . 263

xvii

List of Tables

Table 3.1 Nominal properties of a Fiberite E-glass/934 epoxy lamina (50–55%

fiber volume). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Table 3.2 Mechanical properties of Fiberite E-glass/934 epoxy material system in

principal material coordinates.. . . . . . . . . . . . . . . . . . . . . . . 42

Table 3.3 Fiberite E-glass/934 epoxy CTE in principal material directions as a

function of temperature. .. . . . . . . . . . . . . . . . . . . . . . . . . 46

Table 3.4 Comparison of measured and predicted extensional stiffnesses for a

Fiberite E-glass/934 epoxy quasi-isotropic (45/0/-45/90)2s laminate. . 48

Table 3.5 Summary of the thermomechanical properties of Fiberite E-glass/934

epoxy material system in principal material coordinates.. . . . . . . . 49

Table 3.6 Transformation temperatures for the provided Nitinol alloy in a

fully-annealed state.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Table 3.7 Summary of Nitinol characterization tests.. . . . . . . . . . . . . . . . 61

Table 3.8 Summary of the thermomechanical properties of the Nitinol ribbon. . 68

Table 5.1 Material properties for aluminum specimens.. . . . . . . . . . . . . 114

Table 5.2 Material properties in principal material coordinates for a

graphite-epoxy lamina. . .. . . . . . . . . . . . . . . . . . . . . . . . 117

Table 6.1 Material properties for aluminum specimens.. . . . . . . . . . . . . 141

Table 6.2 Comparison of aluminum 0.4572x0.0254x0.0023 m (18x1x0.090 inch)

simply-supported beam modal frequencies as predicted by classical

and finite element analyses.. . . . . . . . . . . . . . . . . . . . . . . 141

xviii

Table 6.3 Comparison of mid–span RMS responses for an aluminum

0.4572x0.0254x0.0023 m (18x1x0.090 inch) simply-supported beam as

predicted by classical and finite element analyses. .. . . . . . . . . 142

Table 6.4 Comparison of aluminum 0.3556x0.254x0.001 m (14x10x0.040 inch)

simply-supported panel modal frequencies as predicted by classical

and finite element analyses.. . . . . . . . . . . . . . . . . . . . . . . 144

Table 6.5 Comparison of center-point RMS responses for an aluminum

0.3556x0.254x0.001 m (14x10x0.040 inch) simply-supported panel as

predicted by classical and finite element analyses. .. . . . . . . . . 145

Table 6.6 Material properties for aluminum beam specimen in dynamic response

test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

Table 6.7 Material properties for glass–epoxy beam specimen in dynamic

response test.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

Table 6.8 Summary of RMS measured displacements at position 2 on the

SMAHC beam and modal parameters versus temperature.. . . . . . 164

Table 7.1 Material properties for aluminum panel specimen. .. . . . . . . . . 194

Table 7.2 Receiving microphone array measurement locations, shown

schematically in Figure 6.26.. . . . . . . . . . . . . . . . . . . . . . . 202

Table 8.1 Material properties for glass-epoxy and Nitinol. . . .. . . . . . . . . 207

Table 8.2 Results from baseline, SMAHC, and panels with conventional

response abatement treatments.. . . . . . . . . . . . . . . . . . . . . 222

xix

Chapter 1Introduction

1.1 Background and Motivation

The effects of noise and vibration transmission and propagation proliferate throughout our

everyday experiences. These effects range from annoyance or interference with other activities

to mechanical failures, loss of hearing, and even loss of life. Attempts to attenuate or otherwise

remedy noise and vibration problems date back as far as our acknowledgment of them. Methods

of noise/vibration reduction typically involve increases in one or more of three things; mass,

stiffness, or damping. Conventional noise/vibration abatement attempts typically consist of

post-design treatments to reduce the noise/vibration at the source or reduce the perception of

the noise/vibration. Examples of conventional treatments include sound-absorbing blanket or

foam, lead-vinyl or viscoelastic damping layers, viscoelastic vibration isolators, tuned vibration

absorbers, etc. These add-on treatments can be adequate for some applications where weight is

not an issue and system parameters do not vary widely. However, there are other applications

where weight-efficiency and adaptability are all-important. One class of such applications arises

in the area of high-performance aerospace structures.

Excitation levels for many aerospace vehicle structures are high due to engine noise and

turbulent boundary layer fluctuating pressures. The ever-increasing need for weight-efficient

structures in these applications leads to significant sonic fatigue and interior noise issues. Con-

ventional aerospace structures typically employ passive treatments such as structural stiffening,

constrained-layer damping, and acoustic absorption materials to reduce the structural response

1

and interior acoustic levels. These treatments typically suffer from substantial weight penalty

and are often limited to relatively low temperatures and/or high frequencies. Smart materials

may enable new structural design paradigms for high-performance, durable structures.

A significant amount of research in the last several years has greatly advanced the field of

control of flexible structures through the use of smart materials. The field of smart structures

involves a broad range of technologies that enable the realization of structural systems that are

able to sense and to control their own behavior, such that the range of operational performance

may be extended over conventional structures. Smart structures have the ability to change

stiffness, shape, natural frequency, damping, and other mechanical characteristics in response

to changes in temperature, electric field, or magnetic field. The most common smart materials

are piezoelectric materials, shape memory alloys, magnetostrictive materials, electrorheological

fluids and magnetorheological fluids. All of these adaptive materials have great potential in a

variety of applications. Shape memory alloys (SMAs) appear to have significant advantages for

high-performance aerospace vehicles in the areas of structural response control, structural shape

control, and damping enhancement.

Shape memory alloys have been investigated for a variety of applications since their

discovery. Early potential for applications in self-erecting structures, thermally actuated devices,

and energy-conversion systems was identified by Buehler and Wang [1]. Other potential

applications including damping or energy absorbing devices, thermally actuated couplings and

fasteners, and biomedical devices were briefly described by Wayman and Shimizu [2]. Later,

Wayman [3] and Otsuka and Shimizu [4] gave more detailed descriptions of specific devices such

as pipe couplings, thermostats, a robot hand, and various biomedical systems. Other researchers

have reported progress on similar applications, but an entire new field of applications was

created when Rogers and Robertshaw [5] introduced the idea of embedding SMA actuators in

a composite laminate. A structure of this type has been termed a shape memory alloy hybrid

2

composite (SMAHC).

Numerous researchers have investigated the use of SMA actuators, either external to a

structure or embedded within the structure, for static and dynamic structural control. Studies

have considered SMA actuators, external to the structure, for active vibration control [6, 7, 8]

and shape control [9, 10] of cantilevered beams. Other studies have proposed SMA actuators for

vibration control of space structures [11] and presented analytical formulations to demonstrate

the vibration/structural acoustic control of SMAHC panels [12, 13]. Active vibration control

of flexible linkage mechanisms using SMAHCs has been investigated by Venkatesh et al. [14].

Studies of thermal post-buckling [15] and random dynamic response [16] suppression of SMAHC

panels were presented using a thermoelastic finite element (FE) formulation. A different approach

was offered by Ro and Baz [17, 18, 19], where thermal, static, and dynamic analyses were

developed for the case of SMA actuators passing through sleeves in composite plates. Birman

[20] investigated optimal distributions of SMA actuators in sleeves to improve the buckling

performance of sandwich panels under mechanical loading. Theses have also been devoted to

studying the dynamic response tuning capability of Nitinol actuators [21] and predicting the

structural acoustic behavior of SMAHC panels [22].

Shape memory alloys are a class of materials that exhibit a diffusionless martensitic transfor-

mation when cooled from the higher-temperature austenitic state. The interaction of temperature

and applied stress in driving the martensitic and reverse transformations can be used to exploit

phenomena such as theshape memory effect(SME) [23] andpseudo-elasticity. The SME can be

described in simple terms in the following manner. A SMA can be easily deformed in the low

temperature martensitic condition and can be returned to its original configuration by heating

through the reverse transformation temperature range. This type of SME is termedfree recov-

ery. Conversely, in a constrained recovery configuration, the SMA element is prevented from

recovering the initial strain and a large tensile stress (recovery stress) is induced. A situation in

3

which the actuator performs work (deforms under load) is calledrestrained recovery. Extensive

work has been done to characterize these materials, both qualitatively through theoretical models

[24, 25, 26, 27] and quantitatively for particular alloy compositions [28, 29, 30]. Compilations

of papers have also been published [31, 32, 33, 34], which give a good overview of the char-

acteristics of SMAs and their applications. Birman [35] gave a comprehensive review of work

done in the areas of alloy characterization, constitutive modeling, and applications. Some detail

on thermomechanical characteristics of SMAs will be given in Chapters 2 and 3.

Although there are several alloy systems that exhibit the shape memory effect, Nitinol

has the most engineering significance. The name for this family of alloys was derived from

Nickel, Titanium, and Naval Ordnance Laboratory and was patented by Buehler and Wiley [23].

The SME for this family of alloys is limited to nearly equiatomic compositions, i.e., Nickel

content from 53 to 57 percent by weight. Nitinol is considered to be the shape memory alloy

with the most potential for engineering applications because of its ductility at low temperature,

high degree of shape-recovery capability, large pseudoelastic hysteresis, corrosion and fatigue

resistance, biomedical compatibility, and relatively high electrical resistance.

The literature referred to in this introduction is by no means comprehensive, but should

serve as a guide to work that has been done and the state of the art in this area. Other

references to specific works in the literature appear at relevant places throughout the remainder

of this document. The scope and objectives of the present work, including a description of the

contributions made to the state of the art, are given in the next section.

1.2 Scope and Objectives

A Nitinol alloy was chosen for this study because it is the most readily available alloy

and because of the favorable shape memory and electrical resistance characteristics. The

relatively high electrical resistance is beneficial for experimental situations involving activation

4

of the actuators by resistive heating. Because of the ability of SMA actuators to induce

large tensile forces and recover large strains, they are particularly attractive for applications

involving adaptive structural stiffening (constrained recovery) and shape control (restrained

recovery). The environment associated with many aerospace applications includes inherently

elevated temperatures and broadband surface pressure excitation. Embedded SMA elements are

particularly suited to such applications because they generate global structural stiffening and

obviate control electronics and auxiliary power. It is for these reasons that the present work

will focus on SMAHC panel-type structures with embedded actuators operating in a constrained

recovery mode for adaptive structural stiffening.

The purpose of the present work is to develop and experimentally validate an analytical

tool that captures the material nonlinearity of SMAs and complex thermomechanical behavior

of SMAHC laminates. Constitutive theories for SMAHC structures could be developed from

the previously mentioned models for SMA thermomechanical behavior [24, 25, 26, 27], as was

done by Boyd and Lagoudas [36] and Birman [37]. However, these models are difficult to

use in practice. An alternative approach is to employ a constitutive model which makes use

of experimental measurements of fundamental engineering properties. Two such models are

developed as part of the present work. Differences between the two models and reasons for

development of the refined model are discussed in Chapter 2. The model is used in mechanics-

of-materials approach to develop the constitutive relations for SMAHC material systems. The

model captures the material nonlinearity of the SMA, as well as that for the matrix material

when appropriate, and is shown to be quite general in that it can be used to model constrained,

restrained, or free recovery behavior.

The complex nature of SMAHC structures and the goal of experimental validation dictate the

use of the FE method to facilitate accurate thermomechanical analysis. Because no commercial

FE codes have the capability of modeling the constitutive behavior of SMAHC laminates, a

5

research FE code was developed as part of this work. However, it is a goal of this work to

cast the constitutive model and thermomechanical FE formulation such that it can be readily

implemented in a commercial FE code.

The experimental configurations that will be considered for demonstration of adaptive

structural stiffening by thermal activation of embedded SMA actuators involve immovable in-

plane boundary conditions. This condition leads to a potential for structural instability due

to thermal buckling and demands that the FE analysis be capable of predicting static, thermal

buckling and post-buckling behavior in addition to analysis of the response due to practical

dynamic excitation types. To this end, the constitutive relations are used in conjunction with von

Karman strain-displacement relations to allow for the generality of large, out-of-plane deflection

due to thermal post-buckling.

A brief description of the work performed in this study and contributions made have been

given in this section. The manner in which the detailed discussions of various topics are organized

in this document is given in the following section.

1.3 Research Organization

Some detail of the general thermomechanical behavior of SMAs is given in Chapter 2 with

an emphasis on phenomena observed in the mechanical testing performed in this work. That

is followed by development of a constitutive model to describe the behavior of SMA actuators

embedded in a composite matrix. The constitutive model is used with the von K´arman strain-

displacement relations and classical lamination theory to develop the resultant force and moment

relations for a SMAHC laminate. The particular material system used in this study is described

in Chapter 3. An overview of the processes involved in fabricating SMA hybrid composite

specimens is followed by results from thermomechanical property testing of the constituent

materials.

6

Chapter 4 begins with a description of the rectangular plate element used in the finite element

formulation. This is followed by development of the equations governing the thermomechanical

response of a SMAHC laminate to steady-state thermal and dynamic mechanical loads. The

governing equations are separated into static and dynamic components by assuming small (geo-

metrically linear) dynamic motion about a potentially large (geometrically nonlinear) thermally

deflected shape.

Description of the solution procedures begins in Chapter 5 with the static response. Equations

governing static stability and large static deflection are first developed and then used to present

thermal buckling and thermal post-buckling solution procedures. A overview flowchart is shown

in each case. The solution procedures are then validated against a series of known solutions.

The chapter is concluded with experimental validation of the model for the case of a SMAHC

beam clamped at both ends and heated uniformly by DC electrical current.

Continuous and discrete frequency domain solutions are shown in Chapter 6 for predicting

the dynamic response. The dynamic response solution procedure is validated against known

solutions and experimental measurements for isotropic and conventional composite beam spec-

imens excited by random inertial loading. The analysis is then validated with experimental

measurements of the dynamic response performance of a SMAHC beam specimen subjected

to steady-state thermal and dynamic inertial loads. Chapter 6 concludes with correlation of

predicted and measured dynamic responses of a clamped aluminum panel subjected to random

acoustic pressure, which introduces the topic of acoustic radiation in preparation for Chapter 7.

Continuous and discrete frequency domain solutions are shown in Chapter 7 for predicting

the acoustic radiation from a known panel response. The acoustic radiation solution procedure is

validated against a classical analysis method. Chapter 7 is concluded with comparisons against

experimental measurement from a fully clamped aluminum panel subjected to spatially uniform,

temporally random acoustic pressure.

7

A variety of application and demonstrations are discussed in Chapter 8 which show the

complex thermomechanical behavior of SMAHC structures and the effectiveness of embedded

SMA actuators in attenuating dynamic response as compared to conventional response abatement

approaches. Finally, a detailed summary and concluding remarks are given in Chapter 9.

8

Chapter 2Background on SMAs and Constitutive Modeling

A general discussion of the thermomechanical behavior of SMAs begins this chapter. This

discussion will serve as an overview and motivation of the SMAHC constitutive modeling

section. Material property data for the material system used in this study will be presented in

the next chapter, which will be used to quantify the constitutive model for the specific materials

used. The constitutive model will be used in Chapter 4 to develop a finite element formulation

for the thermomechanical response of SMAHC structures.

2.1 Background on Shape Memory Alloys

A very brief introduction to SMAs and the SME was given in Chapter 1. An overview of

some essential characteristics of SMAs will given in this section to facilitate understanding of

results and discussions to be presented later in this document.

The thermoelastic martensitic and reverse transformations are shown schematically in Figure

2.1 in terms of the martensitic fraction versus temperature. At high temperature, a shape memory

alloy exists in a usually cubic phase called austenite. Martensite is the low-temperature phase

and has a lesser symmetry. The reference temperatures indicated in Figure 2.1 mark the start and

finish temperatures for the forward transformation to martensite and the reverse transformation

to austenite, respectively. The thermoelastic martensitic transformation occurs when the alloy in

the austenitic parent phase is cooled through the transformation temperature rangeMs ! Mf ,

producing martensite. The reverse transformation occurs when the alloy is heated through the

9

Temperature

%M

arte

nsi

te

0

100

Mf Ms AsAf

Ms-martensite startMf-martensite finishAs-austenite startAf-austenite finish

Figure 2.1 Schematic of martensitic and reverse transformations, after Liang and Rogers [13].

reverse transformation temperature rangeAs ! Af and the material structure returns to that of

austenite. It is known that a shape memory alloy specimen shows no macroscopic deformation

during the martensitic transformation, described by Wayman and Duerig in [32] for example.

Because the martensite is of a lower-order symmetry than the parent phase, the crystalline

structure change in the forward transformation produces several martensite variants that are self-

accommodating (calledtwinning) so that no macroscopic deformation results. However, the

crystalline change to austenite is unique. When a stress is applied to the twinned martensite,

10

the variant in the sense of the stress grows at the expense of other variants (calleddetwinning).

Thus, inducing the crystallographically unique reverse transformation causes a strain recovery,

and the material reverts to its original configuration. This sequence is the basis of the SME and

is shown schematically in Figure 2.2.

Austenite

Martensite (twinned)

Martensite (deformed)

Coo

lD

eform

Heat

Figure 2.2 Schematic of the shape memory effect, after Wayman and Duerig in [32].

In Nitinol for example, strains as high as 8% can be completely recovered by heating the

alloy aboveAf (Cross et al. [28]). Recall that this type of SME phenomenon is termed

free recovery. This study is primarily concerned with constrained recovery, where the SMA

11

Temperature, oC

Rec

over

yS

tres

s,M

Pa

0 50 100 150 2000

50

100

150

200

250

300

350

400

3%

4%

5%Note: Heat rate = 22oC per minute

InitialStrain

As

Figure 2.3 Nitinol recovery stress versus temperature and initial strain, after Cross et al. [28].

element is prevented from recovering the initial strain and a large recovery stress is induced.

The recovery stress is a function of initial strain and temperature, as shown in Figure 2.3 for

example. The recovery stress will approach, but obviously not exceed the austenitic yield

stress due to plastic slip or production of stress induced martensite, which will be discussed

subsequently. The relationship between these two yield stresses will be discussed subsequently.

For example, Cross et al. [28] found that the maximum yield stress for an alloy in a 100 mil

diameter rod form was 90 ksi, while the corresponding maximum recovery stress was 80 ksi

at 8% initial strain. The austenitic yield stress is approximately a factor of 10 greater than

that of the corresponding martensitic phase. The Young’s modulus also increases by a factor

of three to four under the same conditions (see Figure 2.4). Note that following a process of

free recovery, no shape change is produced in the alloy when it is cooled again belowMf and

12

Temperature, oC

Mo

dulu

s,G

Pa

0 50 100 150 20020

30

40

50

60

70

80

90

Note: Heating

Martensite

Austenite

Figure 2.4 Nitinol Young’s modulus versus temperature, after Cross et al. [28].

the specimen must be strained again to repeat the SME. This type of behavior is termedone-

way shape memory.Two-wayshape memory, where the alloy “remembers” a low-temperature

martensitic configuration and a high-temperature austenitic configuration, can be imparted to

a SMA with appropriate training, as described by Perkins and Hodgson in [32]. One-way or

two-way shape memory behavior can betrained (rendered stable and predictable) into a SMA

by a number of means. A description of one simple method follows.

One-way training can be accomplished by imparting a specific amount of plastic (detwinning)

deformation to the SMA in the low-temperature martensitic condition, heating the material to

induce the austenite phase transformation and allowing the alloy to recover the initial strain, and

cooling the material to induce the transformation to martensite. This procedure is repeated until

the hysteresis becomes stable. Two-way shape memory training can be performed by restraining

13

the SMA element, after being plastically deformed at low temperature, and subjecting it to a

number of thermal cycles, alternating between a specific low and high temperature. The benefit

of this training method is that the shape memory element “memorizes” two stable shapes (a

low-temperature shape and a high-temperature shape) at the expense of some degradation in

performance.

There are other means by which a one-way actuator can be returned to a strained state

following recovery. Two examples include operating actuators in agonist-antagonist pairs or

opposing the actuators with a bias spring. In either case, the active actuators perform work

(deform under load). Recall that operation of an actuator in this type of configuration is

termed restrained recovery. Obviously, the amount of force generated in restrained recovery

is diminished relative to constrained recovery.

The martensitic transformation can also be stress induced in a shape memory alloy at

temperatures aboveMs. An increase in the applied stress produces an effect analogous to

a decrease in temperature. However, only one martensite variant (stress induced martensite,

SIM) grows in the direction most favorable for the applied stress. This phenomenon is

exemplified in the stress-strain curves for an alloy at various temperatures shown in Figure

2.5. For temperatures in the rangeMs < T < As, SIM will be formed and retained after

unloading. Partial reversion to austenite will occur upon stress release for temperatures in the

rangeAs < T < Af . At temperatures approaching or greater thanAf , the complete reverse

transformation ensues immediately upon stress release as the martensite is completely unstable

without stress assistance. These phenomena are all classified as pseudoelastic effects. The

latter effect of full reverse transformation upon stress release is an example of a subset of

pseudoelasticity called superelasticity.

The pseudoelasticity discussed here is called transformation pseudoelasticity. Twinning

pseudoelasticity exists as well, which gives rise torubber-like behavior. This phenomenon

14

Figure 2.5 Stress-strain curves for a shape memoryalloy at various temperatures, after Funakubo [33].

occurs when a SMA is allowed to age in the martensitic phase and is then stressed to induce

prestrain. Instead of maintaining the strain, the strain is recovered upon stress release (in a very

similar manner to superelasticity, but entirely in the martensitic phase). This behavior indicates

that if one is going to prestrain a SMA actuator, it must be done shortly after inducing the

martensitic transformation on cooling, i.e., immediately after removing strain induced in the

material during packaging.

Additional insights can be afforded from the stress strain curves shown in Figure 2.5. When

the material is in the purely martensitic phase, the yield stress to induce martensite detwinning

15

Figure 2.6 Stress-strain curves for a shape memory alloy atvarious temperatures, after Wayman and Duerig in [32].

and produce appreciable strains is quite small. If loading were continued as shown in Figure

2.6, the martensite would eventually become completely detwinned and would deform elastically

until plastic slip is induced. Obviously, the amount of reversible strain is limited by the onset

of plastic slip in the martensitic phase. Rather complex behavior is involved at intermediate

temperaturesMf < T < Af depending upon the state of the alloy. The stress required to yield

the material in the austenitic phase is much higher than that for martensite. At temperatures

in the vicinity of Af , yielding is due to the production of SIM. The stress required to induce

SIM continues to increase with increasing temperature aboveAf until a critical temperature is

reached, at which point plastic slip will be induced in lieu of SIM.

The transformation temperatures of an alloy can be measured by monitoring practically any

property of the alloy, such as electrical resistance, but a particularly accurate and insightful

means is differential scanning calorimetry (DSC). The transformation temperatures for Nitinol

can be altered by changing the composition of the alloy. For example, Buehler and Wang [1]

16

Nickel, weight %

Aus

teni

teS

tart

Tem

pera

ture

(As)

,o C

53.5 54 54.5 55 55.5 56 56.5-100

-50

0

50

100

150

200

Figure 2.7 Variation in austenite start temperature withweight % Ni in Nitinol, after Buehler and Wang [1].

report thatAs for Nitinol can be varied from –50�C to 166�C through changes in the Nickel

content from 54.5% to 56.5% by weight (see Figure 2.7). The transformation temperatures are

also strongly influenced by stress. It is known from early work, such as Cross et al. [28] for

example, that the best shape recovery performance for fully annealed Nitinol is achieved with an

anneal temperature of approximately 930�F. Goldstein et al. [38] found that partial annealing

17

of the residual stresses due to cold work can be used to change the transformation temperatures

and induce a rhombohedralR phase.

All of the discussions to this point have assumed that the alloy is in a fully annealed state.

Some annealing is necessary to remove some of the cold work dislocations which impede the

twin boundary movement, but it is usually desirable to only partially anneal the alloy and leave

some of the dislocations for strength and durability. Production of the R-phase during cooling

from austenite is a direct result of partial annealing, as observed by Saburi in [34]. The R-phase

is itself a product of a thermoelastic martensitic transformation and may be of interest for some

applications because of its lesser hysteresis and reversible strain. The R-phase is simply a by-

product of circumstances for the focus of the work presented here and does not significantly

affect the performance of alloy. The effects of the R-phase are measurably manifested in all

aspects of the alloy characteristics, however. For example, stress-strain curves may show an

additional knee due to R-phase rearrangement under certain conditions, and DSC plots exhibit

an additional endothermic peak due to the formation the R-phase.

The Nitinol actuator material used in this study is partially annealed in the as-received

condition, so the material will exhibit the presence of R-phase. The actuators will be used in a

one-way mode of constrained recovery operation, but characterization of the dependency of the

material on thermal cycle will be quantified in lieu of one-way training. The Nitinol actuators

will be embedded within a composite matrix, along the direction of the matrix fibers, and bonded

within the matrix during cure. Constrained recovery operation will be insured by mechanical

constraint on the resulting SMAHC specimens.

2.2 Constitutive Modeling

A constitutive model of the SMAHC material system is needed in order to predict the

thermomechanical behavior of structures manufactured from the SMAHC material. Constitutive

18

theories for SMAHC materials could be developed from micro-structural shape memory models

(e.g., [24, 25, 26, 27]), as was done by Boyd and Lagoudas [36] and Birman [37]. However,

these models are difficult to use in practice because they are qualitative in nature and depend

upon determination of impractical parameters/material properties. An alternative approach is to

employ a constitutive model which makes use of experimental measurement of basic engineering

properties. Two such models were developed as part of the present study and will be discussed in

this section. The models are developed by a mechanics-of-materials approach (see Jones [39]).

A representative volume element of a SMAHC lamina is shown in Figure 2.8. The element

is taken to be in the plane of the plate. Note that the matrix material is a composite material

itself, for example graphite-epoxy, with principal material directions 1 and 2, and the SMA

actuator is embedded in the 1–direction.

σ2

σ1

Composite Matrix

Composite Matrix

SMA Fiber

2

1

Figure 2.8 Representative volume element for a SMAHC lamina.

Consider the uniaxial state of stress in a SMA actuator along the actuator axis. One form for

the thermoelastic constitutive relation of the SMA actuator in the 1–direction was proposed by Jia

19

and Rogers [5] and was extended to include thermal expansion by Turner, Zhong, and Mei [16]:

�1a = Ea�1 + �r T � As

�1a = Ea(�1 � �1a�T ) T < As

(2.1)

whereEa is the Young’s modulus of the SMA,�1 is the strain in the 1–direction,�r is the recovery

stress of the activated SMA, and�1a is the thermal expansion coefficient for the inactivated SMA.

Note that the Young’s modulus and recovery stress of the SMA are functions of temperature.

The recovery stress is also a function of the initial strain. The thermal expansion coefficient can

be treated as essentially constant for temperatures below the phase transformation temperature.

A detailed derivation of the constitutive relations for a SMAHC lamina using this approach,

referred to as the recovery stress model (RSM), is given in Appendix A. A complete finite element

formulation was developed by Turner, Zhong, and Mei [16] for the thermomechanical response

of SMA hybrid composite panel-type structures based upon the RSM constitutive relations. The

RSM constitutive relations and the resulting finite element governing equations are in a form that

is physically intuitive, but not consistent with conventional thermoelastic analysis and, therefore,

not easily adapted to commercial finite element codes. It will also be shown that the RSM

misses some of the mechanics of the material system. Thus, the constitutive relations will be

re-cast in a more familiar and robust form.

The uniaxial thermoelastic constitutive relation for the SMA can be written in terms of an

effective coefficient of thermal expansion (CTE):

�1a = Ea

h�1 �

TZ

To

�1a(� )d�i

(2.2)

The way in which this relation is used depends upon the application. A measure of the nonlinear

CTE �1a(T ) over the temperature range of concern would be appropriate for free or restrained

recovery applications. For constrained recovery applications, however, one must resort to

measurement of the recovery stress.

20

Comparison of equation (2.2) with equations (2.1) shows the functionality of the effective

CTE for constrained recovery applications; it has the usual effect for temperatures below the

austenite start temperature

TZ

T0

�1a(� )d� = �1a�T (2.3)

and is related to the recovery stress at temperatures aboveAs by the equation

�r = �Ea

TZ

T0

�1a(� )d� or

TZ

T0

�1a(� )d� = ��r

Ea

(2.4)

Note that in this case, the nonlinear thermoelastic nature of the SMA is still captured, albeit in

a different way, because measurements of recovery stress and modulus versus temperature are

inherently cumulative (integrated). Also, note that an experimental approach could be devised

to measure appropriate recovery stress�r quantities for restrained recovery applications as an

alternative to direct measurement of�1a(T ).

Thus, the general form for the thermoelastic constitutive relation in the 1-direction for a

prestrained SMA actuator can be written as

�1a = Ea

h�1 �

TZ

T0

�1a(� )d�i

and sgn(�1a) =n+1 T < As

�1 T � As

(2.5)

where�1a is due to thermal expansion only forT < As and is determined by direct measurement

or from equation (2.4) forT � As.

In order to derive the thermoelastic constitutive relation for the 1–direction of a SMA hybrid

composite, it is assumed that a stress�1 acts alone on the element shown in Figure 2.8(�2 = 0)

and that the SMA and composite matrix are strained by the same amount,�1 (i.e., plane sections

remain plane). The 1-direction thermoelastic constitutive relation of the SMA actuator is given

by equation (2.5). Similarly, the one-dimensional, stress-strain relation in the 1-direction for the

21

composite matrix can be expressed as

�1m = E1m

h�1 �

TZ

T0

�1m(� )d�i

(2.6)

Temperature dependence of the matrix thermal expansion coefficient has been included for

generality.

The resultant force in the 1-direction(�2 = 0) is distributed over the SMA and composite

matrix and can be written as

�1A1 = �1aAa + �1mAm (2.7)

where (�1; A1), (�1a; Aa), and (�1m; A1m) are the (stress, cross-sectional area) of the entire

element, SMA actuator, and composite matrix, respectively. Thus, the average stress�1 is

�1 = �1ava + �1mvm (2.8)

whereva = Aa=A1 and vm = Am=A1 are the volume fractions of SMA and composite matrix,

respectively. Substituting for�1, �1a, and�1m results in

E1

h�1 �

TZ

T0

�1(� )d�i= Eava

h�1 �

TZ

T0

�1a(� )d�i+ E1mvm

h�1 �

TZ

T0

�1m(� )d�i

(2.9)

So, the Young’s modulus and thermal expansion coefficient for the hybrid composite in the

1–direction have the form

E1 = Eava + E1mvm (2.10)

and

TZ

T0

�1(� )d� =

EavaTRT0

�1a(� )d� + E1mvmTRT0

�1m(� )d�

Eava + E1mvmsgn(�1a) =

n+1 T < As�1 T � As

(2.11)

22

A similar constitutive relation may be derived for the 2–direction by assuming that the

applied stress�2 acts upon both the SMA and the matrix(�1 = 0). The uniaxial thermoelastic

constitutive relations in the 2–direction for the SMA actuator and the composite matrix are

analogous to those in equations (2.2) and (2.6), i.e.,

�2a = �2 = Ea

h�2a �

TZ

T0

�2a(� )d�i

(2.12)

and

�2m = �2 = E2m

h�2m �

TZ

T0

�2m(� )d�i

(2.13)

respectively. The CTE in equation (2.12) for the SMA does not depend on the recovery stress,

but is still nonlinear due to the differing martensitic and austenitic properties.

The total elongation is due to strain in the composite matrix and the SMA actuator and may

be written in the form

A1�2 = Am�2m +Aa�2a (2.14)

Thus, the total strain becomes

�2 = �2mvm + �2ava (2.15)

Substituting for�2, �2a, and �2m results in

�2

E2

+

TZ

T0

�2(� )d� =�2va

Ea+�2vm

E2m

+

TZ

T0

[�2a(� )va + �2m(� )vm]d� (2.16)

Therefore, the Young’s modulus and thermal expansion coefficient for the hybrid composite in

the 2–direction have the form

E2 =E2mEa

(E2mva + Eavm)(2.17)

andTZ

T0

�2(� )d� =

TZ

T0

[�2a(� )va + �2m(� )vm]d� (2.18)

23

Expressions for the hybrid composite Poisson’s ratios and shear moduli follow from similar

derivations:

�12 = �ava + �12mvm G12 =GaG12m

Gavm +G12mva(2.19)

The thermoelastic constitutive relations in principal material coordinates for an orthotropic

material have the following formf�g = f�g

M+ f�g

T

= [D]f�g+

TZ

T0

f�(� )gd�(2.20)

wheref�g is the total strain,f�gM

is the mechanical strain,f�gT

is the thermal strain, and[D]

is the compliance matrix. These relations can be inverted to give

f�g = [C]f�g � [C]

TZ

T0

f�(� )gd� (2.21)

where[C] is the stiffness matrix. The constitutive relations, for plane stress, reduce to8<:�1�2�12

9=; =

24Q11 Q12 0Q12 Q22 0

0 0 Q66

358<:�1�2 12

9=; �

24Q11 Q12 0Q12 Q22 0

0 0 Q66

35

TZ

T0

8<:�1(� )�2(� )

0

9=;d� (2.22)

where [Q] is the reduced stiffness matrix of the composite lamina. The reduced stiffnesses are

related to the engineering properties through the following equations

Q11 =E1

1� �12�21Q12 =

�12E2

1 � �12�21Q22 =

E2

1 � �12�21Q66 = G12 (2.23)

The expressions for the hybrid lamina engineering properties in terms of the properties of the

constituent materials are reiterated here for convenience.

E1 = Eava + E1mvm E2 =EaE2m

Eavm + E2mva

�12 = �ava + �12mvm G12 =GaG12m

Gavm +G12mva

TZ

T0

�1(� )d� =

Eava

TRT0

�1s(� )d� + E1mvm

TRT0

�1m(� )d�

Eava + E1mvmsgn(�1a) =

n+1 T < As

�1 T � As

TZ

T0

�2(� )d� =

TZ

T0

[�2a(� )va + �2m(� )vm]d�

(2.24)

24

The thermoelastic constitutive relations for a general k-th orthotropic layer of a laminate

with an arbitrary fiber/SMA actuator orientation angle�, relative to the 1–direction in Figure

2.8, follow from transformation of the relations in principal material coordinates, equation (2.22).8<:�x�y�xy

9=;k

=�Q�k

8<:

�x�y xy

9=;k

�

�Q�k

TZ

T0

8<:

�x(� )�y(� )

�xy(� )

9=;k

d� (2.25)

or

f�gk =��Q�k(f�gk � f�gTk) (2.26)

The transformed reduced stiffness matrix��Q�k

is related to the reduced stiffness matrix in

principal material coordinates by

��Q�k= [T ]

�1[Q]k[T ]

�T (2.27)

and [T ] is the transformation matrix given by

[T ] =

24 cos2 � sin2 � 2 sin � cos �

sin2 � cos2 � �2 sin � cos �

�2 sin � cos � 2 sin � cos � cos2 � � sin2 �

35 (2.28)

The constitutive relations given by equations (2.22) or (2.26) will be referred to as the effective

coefficient of thermal expansion model (ECTEM).

The constitutive equations in principal material coordinates, equations (2.22), will now be

considered further to clarify some advantages of the ECTEM approach. The expressions for the

thermal strains in equations (2.24) allow the constitutive relations to be rewritten as8<:�1�2�12

9=; =

24Q11 Q12 0

Q12 Q22 00 0 Q66

358<:

�1�2 12

9=; �

24Q11 Q12 0

Q12 Q22 00 0 Q66

35

TZ

T0

8<:c1a�1a(� )

va�2a(� )0

9=;d�

�

24Q11 Q12 0Q12 Q22 0

0 0 Q66

35

TZ

T0

8<:c1m�1m(� )vm�2m(� )

0

9=;d�

(2.29)

where the thermal effects of the SMA and matrix material have been separated and

c1a =Eava

Eava + E1mvmand c1m =

E1mvm

Eava + E1mvm(2.30)

25

are constants. These constitutive relations have a form similar to those in the RSM [16], which

are rewritten here for convenience of comparison:

8<:�1�2�12

9=; =

24Q11 Q12 0

Q12 Q22 0

0 0 Q66

358<:�1�2 12

9=;+

8<:�r

0

0

9=;va � [Q]

m

8<:�1�20

9=;m

vm4T

= [Q]

8<:�1�2 12

9=;+

8<:�r

0

0

9=;va �

0@[Q]

8<:�1�20

9=;v1Am

4T T > As

(2.31)

While the RSM only allows the recovery stress to affect the stress in the 1-direction of a lamina

(term 2 of equation (2.31)), the analogous term in the ECTEM (term 2 of equation (2.29)) shows

an effect of the 1-direction effective CTE on the stress in the 2-direction of the lamina, through

the reduced stiffnessQ12. The ECTEM also allows for the fully nonlinear functionality of the

2–direction CTE for the SMA while the RSM neglects it altogether forT > As. As a result

of these differences, the thermal forces calculated by the two formulations are different and

the ECTEM more accurately captures the behavior of the material system. The form for the

ECTEM constitutive relations given in equation (2.29) will be useful for reference in the thermal

buckling discussion of Chapter 5.

In summary, the ECTEM essentially reduces to the RSM for constrained recovery applica-

tions, but captures the two-dimensional behavior of SMA actuators and their interaction with

a composite matrix more accurately than the RSM. The ECTEM is also capable of modeling

restrained or free recovery behavior in a similar manner with experimental measurements of

SMA recovery stress, modulus, and/or nonlinear CTE.

2.3 Deformation Behavior and Stress Resultants

A model for the deformation behavior of a SMAHC laminate will be necessary, along with

the constitutive relations, to develop the equations governing the static and dynamic response

of such structures. Recall that one demand on the formulation is to model the thermoelastic

26

response in the presense of immovable in-plane boundaries, i.e., thermal buckling and post-

buckling analysis. Thus, the midplane strain and curvature vectors are defined from the von

Karman strain-displacement relations [40]:

8<:�x�y xy

9=; = f�og+ zf�g

= f�omg+ f�obg+ zf�g

=

8<:

u;xv;y

u;y+v;x

9=;+

1

2

8<:

w;2xw;2y

2w;xw;y

9=;� z

8<:w;xxw;yy2w;xy

9=;

(2.32)

to allow for potential membrane stretching effects due to geometrically nonlinear deflections

associated with thermal post-buckling. Note that the midplane strainf�og consists of two

components; the midplane strain due to membrane displacementsf�omg and the nonlinear

midplane strain due to bending displacements��ob

.

Substitution of the strain-displacement relations in equation (2.26) results in

f�gk =��Q�k(f�og+ zf�g)�

��Q�kf�gTk (2.33)

The resultant force and moment vectors of the SMA hybrid composite laminate are defined as

fN;Mg =

h=2Z�h=2

f�gk (1; z)dz (2.34)

or

�N

M

�=

�A B

B D

��o

�

��

�N�TM�T

�(2.35)

where the laminate stiffness[A]; [B] and [D] and the thermal resultant force and moment vectors

fN�T g andfM�T g are all temperature dependent. The laminate stiffness matrices have the

27

usual form

[A] =

h=2Z

�h=2

��Q�kdz =

NlayXk=1

��Q�k(zk � zk�1)

[B] =

h=2Z

�h=2

��Q�kzdz =

1

2

NlayXk=1

��Q�k

�z2k � z2k�1

�

[D] =

h=2Z

�h=2

��Q�kz2dz =

1

3

NlayXk=1

��Q�k

�z3k � z3k�1

�

(2.36)

and the expression for the thermal resultants is as follows.

fN�T ;M�T g =

h=2Z

�h=2

(1; z)��

�Q�k

TZ

T0

f�(� )gk

�d�dz (2.37)

Expanding the resultant force and moment relations, equation (2.35), in terms of the midplane

strain and curvature components, equation (2.32), results in

fNg = [A]f�omg+ [A]f�obg+ [B]f�g � fN�Tg

= fNmg+ fN�g+ fNbg � fN�T g(2.38)

andfMg = [B]f�omg+ [B]f�obg+ [D]f�g � fM�T g

= fMmg+ fM�g+ fMbg � fM�T g(2.39)

where the subscriptsm, �, b, and�T indicate components induced by membrane strain, nonlinear

membrane strain, bending strain, and thermal expansion, respectively. The resultant force and

moment relations and the strain-displacement relations presented in this section will be useful

for reference in formulating the equations governing the thermomechanical response of SMAHC

panel-type structures in Chapter 4.

A review of the behavior of SMAs was given in this chapter followed by development

of a constitutive model for SMA actuators embedded in a composite matrix and used in a

constrained-recovery model of the SME for adaptive stiffening. In order to use this model to

predict the behavior of the SMAHC material system used in this study, the thermomechanical

28

properties of the constituents must be determined. Results from such characterization testing will

be presented in the next chapter, following motivation of the selected materials, description of the

SMAHC material system configuration, and discussion of the processes involved in fabricating

the SMAHC specimens.

29

Chapter 3Specimen Fabrication and Material System Characterization

A constitutive model was developed in the previous chapter to describe the mechanics of a

SMAHC material system. That model requires measurement of a number of material properties

from the constituent materials in order to quantify the behavior of a structure fabricated from such

a material. In this chapter, a brief motivation is given for the selected material system, followed

by details of the SMAHC specimen fabrication processes and hardware. Processes involved in

and results from thermomechanical tests on the constituent materials will be discussed, and the

chapter will conclude with disclosure of the final material property data set used to generate the

numerical results for SMAHC specimens in this study.

3.1 Material System Selection

In the typical application being considered, such as adaptive stiffening of aerospace struc-

tures, the SMA elements would be activated through an inherently elevated temperature of the

service environment. Thus, activation of the SMA by surface heat in the planned experiments

would be the most accurate simulation. Surface heating could be accomplished through the use

of quartz radiant heaters or a number of other sources. These heating methods involve practical

problems, however, such as the need for thermal insulation of the test article support structure and

other equipment, interference with structural/acoustic/thermal measurements, thickness-direction

asymmetry of the thermal load, and a complicated thermal control system. Alternatively, the

SMA can be activated electrically through resistive heating. This method has the advantages of

30

simple operation, relatively uniform heating, and excellent controllability. Electrical isolation of

the SMA elements may be required for electrical activation of laminates involving carbon-epoxy

matrices. Electrical isolation can be achieved through coating techniques, but this could present

significant complications. It was decided that a glass-epoxy matrix would be used to circumvent

these problems as experiments performed in this study are intended for demonstration and theo-

retical validation only. This material system also affords the advantage of visual flaw detection

due to its translucency.

Most recent SMA hybrid composite structural designs have focused on SMA wire-type

actuators distributed uniformly throughout the laminate or individual laminae. There are several

disadvantages associated with the use of SMA wire actuators. Integration of wire-type actuators

into a composite leads to complicated fabrication procedures because of the relatively large

number of actuators usually needed. The small surface area of the actuators makes them very

sensitive to actuator/matrix interface flaws because voids can be of significant relative size. Small

size has also been blamed for a relatively high rate of actuator breakage during cure because of

sensitivity to stress concentrations at the mechanical restraints. Finally, it is difficult to achieve a

desirable overall volume fraction of SMA when trying to optimize the integration of the actuators

by placing them in only selected layers. Therefore, the present work will focus on SMA hybrid

composite laminates involving ribbon-type SMA elements to simplify the fabrication procedure,

allow for more flexibility in fabrication, and hopefully desensitize the actuators to interface voids

and stress concentrations. For example, the ribbon-type SMA elements can be placed in strips

cut out of particular matrix laminae to result in a rather simple lay-up operation, while keeping

the volume fraction high in the desired locations. Another potential benefit of this type of lay-up

procedure is that it could potentially be extended directly to e-beam consolidation procedures

where spools of SMA ribbon would strategically replace spools of matrix material ribbon.

In summary, the material system used in this study consists of a glass-epoxy matrix material

31

with embedded Nitinol ribbon. Details of the SMAHC specimen fabrication procedures/hardware

are given in the following section, followed by a discussion of the thermomechanical testing of

the constituent materials.

3.2 Specimen Fabrication

Although a panel-type specimen is closer to the configuration of intended applications, a

more simple test specimen will suffice for the objectives of this study; demonstration of the

thermomechanical response of a SMAHC laminate and experimental validation of the theoretical

model developed in this work. A schematic of the selected beam-type specimen is shown in

Figure 3.1. Various constraints, other than the use of ribbon-type actuators, influenced the design

of this laminate. These constraints include fabrication tooling geometry and dynamic range of

anticipated experimental configurations. The specimen design is by no means optimized, but the

predicted performance is in a range that is suitable for the expected loading conditions and to

demonstrate significant improvement from the SMA reinforcement.

Note that this specimen design calls for a SMA actuator width of 0.0127 m (0.5 inch) to be

embedded within each 0� glass-epoxy layer, i.e., replace a portion of each 0� layer. Material

availability limited the procured SMA ribbon cross section to 0.0023x1.5e-4 m (0.09x0.006

inches). This was considered acceptably close to the glass-epoxy unidirectional tape thickness

(�1.27e-4 m,�0.005 inches) and it was planned that five widths of the ribbon would be placed

side-by-side to develop a width near the half-inch in the specimen design. Alternatively, in the

case that the adjacent SMA actuators prove detrimental because of a weak interface between

actuators, the ribbons could be separated within laminae or between laminae.

The SMAHC specimen fabrication procedure used in this study consists of the following

steps; prestrain the SMA actuators, lay-up the composite laminate, cure the laminate while

restraining the actuators, and machine beam specimens from the consolidated SMAHC laminate.

32

22"x1"x0.080" Glass-Epoxy/SMA Hybrid Beam (6 Count)(45/0SMA/-45/90)2sym Lamination

SMA in 0o Laminae Only

Cut Beams from Larger Laminate

0.5"1"

22"

SMA Ribbon:0.5"x0.005" Cross Section6" Overhang at Beam Ends

Figure 3.1 Schematic of beam specimens cut from a SMA hybrid composite laminate.

The Nitinol material was packaged on a spool, which subjected the ribbon to significant out-

of-plane bending strain. Some in-plane bending, axial, and torsional strain was also imparted

to the ribbon during the spooling operation. Thispackagingstrain must be removed prior to

imparting the actuator prestrain. Packaging strain removal can be conveniently done by inducing

the reverse (austenitic) transformation. An apparatus was developed to perform this operation,

shown in Figure 3.2. It is desirable to induce the martensitic transformation just prior to

prestraining in order to avoid aging effects, such as the rubber-like form of pseudoelasticity.

33

DC Power Supply

Ribbon Grips

Bakelite Dovetail Track

Figure 3.2 Packaging strain recovery rig.

Inducing the transformation is also accomplished during the packaging strain recovery process

as the ribbon cools to ambient temperature. It will be shown in the next section that, for this

particular Nitinol composition in an unrestrained condition, the martensitic transformation is not

complete at ambient temperature. However, the martensite will be stress induced during the

prestraining operation. Stress–inducing the martensite results in essentially the same state as

would be achieved by cooling the ribbon to sub-ambient temperatures prior to prestraining. The

main difference is that there may be some austenite remaining in the ribbon after prestraining

by the former method, but should be of no consequence because any subsequent thermal cycling

(e.g., composite cure) will render the state from the two methods equivalent. The former method,

without cooling to sub-ambient temperatures, was selected for simplicity.

The prestraining operation needs to be very precise and repeatable. It was decided that

the actuators would be elongated on a mechanical loading machine operated in stroke-control

34

Alignment Groove

KnurlingMTS Grip Tabs

Figure 3.3 Nitinol ribbon gripping apparatus for prestraining and mechanical testing.

mode. Alignment of the ribbon during elongation is critical to avoid bending strain and to

obtain repeatable results. Handling of the ribbon is cumbersome during installation in the test

machine, and it is desirable to prestrain multiple strands at once to expedite the process. In

order to rectify these issues, an apparatus was developed to align and maintain proper geometry

on three ribbon lengths at once during installation in the test machine. A close-up view of the

alignment/gripping apparatus is shown in Figure 3.3. The entire assembly (alignment/gripping

apparatus, spacer, and three ribbons) is shown installed in the test machine in the left half of

Figure 3.4. The Nitinol ribbons are shown ready for elongation, subsequent to spacer removal,

in the right half of Figure 3.4.

The particular spacer length shown was used to maintain a SMA ribbon gage length of

0.635 m (25 in) for manufacture of the beam specimens shown in Figure 3.1. The gage length

of the ribbons was elongated to a length of 0.6604 m (26 in), which corresponds to a prestrain

of 4%. This gage length was used in order to ensure uniform prestrain over the 0.6604 m (26

in) span between mechanical restraints on the composite lay-up tooling, which will be discussed

subsequently. A total of fifty-four ribbon lengths were prestrained 4% using this procedure to

35

MTS Hydraulic Grip

Nitinol Alignment/Grip Apparatus

Spacer

3 Ribbons

Figure 3.4 Nitinol ribbon prestrain and mechanical testing apparatus.

prepare for SMAHC laminate fabrication.

A mechanical drawing of the lay-up and cure tooling plate is shown in Figure 3.5. This

apparatus was designed to provide a flat surface (± 0.005 inches) and mechanical grips to restrain

the SMA during an autoclave cure procedure. Recall that the beam specimen design calls for a

lamination of (45/0/-45/90)2s with the Nitinol in the 0� layers. This lamination was accomplished

by cutting widths of the 0� layers to size during assembly and filling the void with five ribbon

36

Figure 3.5 Mechanical drawing of the laminate lay-upand cure tooling apparatus (all dimensions in inches).

widths laid side-by-side. A picture of the lay-up procedure at the stage of finishing the first

0� layer is shown in Figure 3.6. Note that Kapton release film, fiberglass breather cloth,

and teflon-coated fiberglass bleeder cloth are layered between the tooling plate surface and the

37

SMA RibbonGlass/Epoxy

Release Cloth

Tooling PlateRelease Cloth

Figure 3.6 Laminate lay-up procedure at the stage of completing layer 2.

Thermocouples

Release Paper

Knurled Grips

Figure 3.7 Completed laminate with cure control thermocouples and NiTi ribbon grips torqued.

first glass-epoxy layer, in that order. The laminate after complete assembly is shown in Figure

3.7, where the cure cycle control thermocouples have been installed near the laminate and the

Nitinol ribbon ends have been secured within the knurled mechanical restraints. The restraints

consisted of three parts (tooling plate surface, a thin spacer with knurling on both sides, and

the top clamping bar) to minimize bending of the actuators between the laminate boundary and

the restraint and to maximize grip for each layer of SMA by minimizing the number of layers

38

Vacuum Port

Perimeter Seal

Figure 3.8 Laminate after vacuum-bagging and applying vacuum.

Figure 3.9 Consolidated SMAHC laminated panel.

gripped together. The mechanical restraint screws were torqued to 11.30 N•m (100 in•lbf). The

release paper was subsequently removed from the top layer of glass-epoxy and the laminate was

topped with bleeder cloth, Kapton film, a caul plate, bleeder cloth, and breather cloth, in that

order. The entire assembly was vacuum bagged as shown in Figure 3.8 and drawn to an internal

pressure of 0.0475 atm. The assembly was subjected to the autoclave cure cycle recommended

by the vendor (Fiberite, Inc.) and the resulting consolidated part is shown in Figure 3.9. Details

of the cure cycle will be discussed in the following section. Two SMAHC beam specimens were

machined from this laminate using the respective centerlines of the embedded Nitinol ribbon.

The beam dimensions are shown in Figure 3.10, where it can be seen that the beam marked

number 3 has been readied for installation in test fixtures. The beams have dimensions of

39

Figure 3.10 SMAHC beam specimens machined from panel.

0.5588x0.0254x0.0019 m (22x1x0.078 in). The SMA leads of the test-prepared beam were

trimmed to an overall length of 0.6604 m (26 in). The overall volume fraction of the SMA

within the glass-epoxy dimensions is approximately 13.8%.

3.3 Material System Characterization

The material system and processes used to fabricate SMAHC beam specimens was described

in the previous sections. In order to predict the thermomechanical behavior of such a structure

using the constitutive model developed in the previous chapter, a number of material properties

from the constituents must be determined. Characterization testing that was performed on the

glass-epoxy matrix material and Nitinol alloy are described in the following sections.

3.3.1 Glass-Epoxy Matrix

Glass-epoxy unidirectional prepreg tape was obtained from Fiberite, Inc. The material

is Hye 8 End/934, an E-glass/934 epoxy resin material system. The resin material is for-

mulated for autoclave processing and the following cure procedure is recommended; heat at

1.1–2.8�C/minute (2–5�F/minute) to 121.1�C (250�F) and hold for 15 minutes, apply 689.5 kPa

(100 psig) pressure and continue to hold at 121.1�C (250�F) for an additional 45 minutes, heat

at 1.1–2.8�C/minute (2–5�F/minute) to 176.7�C (350�F) and hold for 120 minutes, and cool

at 2.8�C/minute (5�F/minute) (maximum) to ambient. The material and processing properties

shown in Table 3.1 were available from the Fiberite Materials Handbook [41]. Although these

40

Table 3.1 Nominal properties of a Fiberite E-glass/934 epoxy lamina (50–55% fiber volume).

Property Room Temperature

Young’s Modulus, E1 41.37-55.16 GPa (6-8 Msi)

Poisson’s Ratio 0.28-0.32

Density 1.919-1.980e+3 kg/m3 (1.795-1.852e-4 lbf s2/in4)

Cure Temperature 176.7�C (350�F)

Glass Transition 193.3�C (380�F)

properties are useful as estimates, thermal expansion and more precise elastic property char-

acterization are needed for accurate prediction of thermomechanical performance. Therefore,

significant thermomechanical testing of the glass-epoxy material system was performed.

Two laminates were fabricated from the glass-epoxy material system to provide specimens

for thermomechanical testing; a unidirectional lay-up (0)20 and an angle-ply lay-up (±45)4. Tab

material was post-cured to the laminates and the resulting structures were machined into uniaxial

tensile specimens with dimensions of 0.2286x0.0254 m (9x1 inches) (gage length of 0.1524 m,

6 inches). The tensile specimens were instrumented with a “T-strain gage” at the mid-span

and tested on a MTS machine. The unidirectional specimens were tested according to ASTM

standard D 3039/D 3039M for estimates of the Young’s moduli and Poisson’s ratios;E1, E2, �12,

and�21. The angle-ply specimens were tested according to ASTM standard D 3518/D 3518M for

estimates of the shear modulusG12. Although the ASTM standards call for modulus estimates

to the determined over strain ranges of�1000–3000�� and�2000–6000��, respectively, the

transverse (90�) uniaxial and±45 angle-ply stress-strain relations showed discernible nonlinearity

over those ranges. The maximum normal strain and shear strain for the test configurations

considered in this study are not expected to exceed 500 and 250��, respectively. Therefore,E2

andG12 estimates were determined from�50–1000�� and�50–500��, respectively, to achieve

more representative estimates with ample headroom. Results from these tests/calculations are

41

Table 3.2 Mechanical properties of Fiberite E-glass/934epoxy material system in principal material coordinates.

Temperature E1 E2 �12 G21�C GPa GPa — GPa

23.9 49.30 20.00 0.29 9.65

37.8 49.16 19.44 0.29 9.24

65.6 48.75 18.20 0.29 8.41

93.3 48.61 16.20 0.29 7.58

121.1 48.54 14.00 0.29 5.58

148.9 48.82 11.38 0.29 3.45

given in Table 3.2.

Specimens for measurement of the coefficient of thermal expansion (CTE) were also ma-

chined from the unidirectional laminate (0)20 described above. Two CTE specimen geometries

were machined for testing by two different methods. Rectangular 0.1524x0.006 m (6x0.25 inch)

specimens were tested according to ASTM standard E 228 at Materials Research and Engineer-

ing, Inc. A summary of those test results is given in Arvidson et al. [42]. Specimens with the

geometry shown in Figure 3.11 were tested by an interferometric dilatometer method at NASA

LaRC. Both quadratic and cubic interpolation of the raw thermal expansion data are used in

practice to generate linear and quadratic estimates of the CTE with temperature, respectively.

Cubic interpolation was employed with the ASTM E 228 test method while quadratic interpola-

tion was used with the interferometer method. The consequence of this difference can be seen

in Figures 3.12 and 3.13, where the longitudinal (fiber direction) and transverse CTE estimates

versus temperature are shown from the two methods. The two methods are in agreement

for the overall trend in the CTE data, but the cubic interpolation of the data collected with the

ASTM standard captures higher order effects. Data resulting from both methods are tabulated in

Table 3.3, but the quadratically varying CTE data from the ASTM E 228 method will be used

in the predictions reported in this study.

42

Figure 3.11 Schematic of the CTE specimen geometry formeasurement by the interferometric dilatometer method.

43

Temperature, oF

CT

E,µ

in/in

/o F

100 150 200 250 3000

1

2

3

4

5

ASTM E228Interferometer

Figure 3.12 Fiberite E-glass/934 epoxy longitudinal(fiber-direction) CTE vs. temperature by two methods.

44

Temperature, oF

CT

E,µ

in/in

/o F

100 150 200 250 3005

6

7

8

9

10

11

12

13

14

15

ASTM E228Interferometer

Figure 3.13 Fiberite E-glass/934 epoxy transverse CTE vs. temperature by two methods.

45

Table 3.3 Fiberite E-glass/934 epoxy CTE in principalmaterial directions as a function of temperature.

�1 �2

Temperature ASTM E228 Interferometer ASTM E228 Interferometer�C ��/�C ��/�C ��/�C ��/�C

23.9 — 6.17 — 21.04

26.7 5.83 6.21 13.81 21.13

37.8 6.95 6.34 21.02 21.49

48.9 7.22 6.44 24.14 21.85

60.0 7.00 6.55 24.66 22.23

71.1 6.62 6.68 23.90 22.59

82.2 6.34 6.79 22.79 22.95

93.3 6.25 6.91 22.00 23.31

104.4 6.39 7.02 21.92 23.67

115.6 6.77 7.13 22.66 24.05

126.7 7.18 — 24.01 —

137.8 7.42 — 25.47 —

Tensile specimens with a geometry of 0.1524x0.0127 m (6x0.5 inches) were machined from

scrap (glass-epoxy-only) material remaining subsequent to machining SMAHC specimens from

the larger quasi-isotropic laminate shown in Figure 3.9. A schematic of the specimen machining

organization from the SMAHC panel is shown in Figure 3.14. The tensile specimens were

tabbed and instrumented with a “T-strain gage” at the mid-span prior to testing. These quasi-

isotropic specimens were tested according to ASTM standard D 3039/D 3039M for estimates

of the extensional stiffnesses;A11 andA22. A brief discussion of these measurements is given

in the following paragraph.

Recall the stress resultant relations given in equation (2.35). In performing the uniaxial tensile

tests on the quasi-isotropic specimens, the applied forceNx, axial strain�x and transverse strain

�y were measured. The transverse and shear force components are zero,Ny = Nxy = 0.

46

Figure 3.14 Schematic of the machining process for removing tensile,CTE, and SMAHC beam specimens from the SMAHC laminate.

For a symmetric quasi-isotropic laminate, the shear-extension coupling stiffnesses are zero,

A16 = A26 = 0 and the extensional stiffnesses are equalA11 = A22. With these simplifications,

the stress resultant equations reduce to

Nx = A11�o

x+A12�

o

y

0 = A12�o

x+A11�

o

y

0 = A66 o

xy

(3.1)

47

Table 3.4 Comparison of measured and predicted extensional stiffnessesfor a Fiberite E-glass/934 epoxy quasi-isotropic (45/0/-45/90)2s laminate.

Extensional Stiffness, A11

Temperature Measured Predicted�C MN/m MN/m

23.9 69.2 63.5

37.8 63.3 62.4

65.6 60.1 60.1

93.3 57.5 57.3

121.1 53.8 53.2

148.9 50.7 48.9

The relationship in the second equationA12 = �A11�o

y=�o

xallows the first equation to be rewritten

as

Nx=�o

x= A11

h1 �

��oy=�o

x

�2ior A11 =

Nx=�o

xh1�

��oy=�o

x

�2i (3.2)

Thus, the measured quantities are sufficient to determine the extensional stiffnesses of the quasi-

isotropic laminate.

The uniaxial and angle-ply properties discussed previously can be used with classical

lamination theory (CLT) to calculate extensional stiffnesses for the quasi-isotropic laminate. The

extensional stiffnesses measured from the quasi-isotropic specimens are shown in comparison

to those predicted with CLT from the uniaxial and angle-ply properties in Table 3.4. It can

be seen that good agreement is achieved. Thus, usage of the properties in Table 3.2 is deemed

appropriate for predictions of the glass-epoxy material system behavior.

A variety of test procedures and results have been presented in this section. A summary of

the properties to be used in the numerical predictions later in this study are shown in Table 3.5.

Note that slight extrapolation of the CTE data and linear interpolation of the CTE and elastic

property data was performed to result in a common temperature resolution.

48

Table 3.5 Summary of the thermomechanical properties of FiberiteE-glass/934 epoxy material system in principal material coordinates.

Temperature E1 E2 �12 G21 �1 �2�C GPa GPa — GPa ��/�C ��/�C

21.1 49.30 20.00 0.29 9.65 5.22 10.80

26.7 49.30 19.86 0.29 9.58 5.83 13.81

37.8 49.16 19.44 0.29 9.24 6.95 21.02

48.9 49.02 18.96 0.29 8.89 7.22 24.14

60.0 48.82 18.48 0.29 8.55 7.00 24.66

65.6 48.75 18.20 0.29 8.41 6.80 24.28

71.1 48.75 17.79 0.29 8.27 6.62 23.90

82.2 48.68 17.03 0.29 7.93 6.34 22.79

93.3 48.61 16.20 0.29 7.58 6.25 22.00

104.4 48.61 15.31 0.29 6.76 6.39 21.92

115.6 48.54 14.41 0.29 6.00 6.77 22.66

121.1 48.54 14.00 0.29 5.58 6.97 23.33

126.7 48.61 13.45 0.29 5.17 7.18 24.01

137.8 48.68 12.41 0.29 4.27 7.42 25.47

148.9 48.82 11.38 0.29 3.45 7.63 26.28

3.3.2 Nitinol Alloy

Nitinol ribbon material was provided by Shape Memory Applications, Inc. A Nitinol alloy

was specified with an appropriate composition to give a martensite finishMf temperature above

the expected ambient temperature while maintaining an austenite finish temperature well below

the glass-transition temperature of the Fiberite 934 epoxy matrix. The achievable ribbon cross

section was 0.0023x0.00015 m (9�0.006 inches). The nominal transformation temperatures of

the provided alloy, in a fully annealed state, were reported by the vendor as shown in Table 3.6.

These temperatures were reportedly determined by differential scanning calorimetry (DSC). As

the provided material is not in a fully annealed state, it was expected that all of the transformation

temperatures would be somewhat lower than those in the table due to dislocations remaining

49

Table 3.6 Transformation temperatures for the provided Nitinol alloy in a fully-annealed state.

Temperature

Transformation Marker �C �F

Austenite Start 67.2 153

Austenite Peak 80.0 176

Austenite Finish 85.0 185

Martensite Start 46.1 115

Martensite Peak 40.0 104

Martensite Finish 32.2 90

from the cold work performed to generate the final geometry. Conversely, mechanical restraint

in the planned test configurations will increase phase transformation temperatures. In any case, it

was decided that additional DSC measurements should be performed on samples of the provided

material to determine the baseline phase transformation temperatures.

Small samples (<10mg) were cut from the ribbon material in the as-received condition for

the DSC testing. Initial scans showed irregularities in the endothermic peak during heating

and a disproportionately small-sized exothermic peak during the cooling, which indicated a

potential for unexpected additional activity below ambient temperature. As a result of these

initial findings, two more-rigorous scanning procedures were defined in an effort to differentiate

between several effects seen in the initial scans. The first scan procedure, termed theheat-first

procedure, entailed heating from ambient temperature to 150�C at 5�C/minute, cooling from

150�C to –20�C at 5�C/minute, and repeating this entire profile once the sample was returned to

ambient from –20�C (also at 5�C/minute). For future reference, the first cycle will be calledfirst

heatand the repeat cycle will be calledsecond heat. The second procedure (cool-first) consisted

of cooling from ambient temperature to –20�C at 5�C/minute, heating from –20�C to 150�C at

5�C/minute, cooling from 150�C to ambient temperature at 5�C/minute, and repeating this entire

profile once the sample was returned to ambient. For this case, the first cycle will be calledfirst

50

Temperature, oC

Uns

ubtr

acte

dH

eatF

low

,mW

0 50 100 15017

18

19

20

21

22

23

24

First HeatSecond Heat

Figure 3.15 Heat-first DSC signatures for first- and second-heat of as-received material.

cool and the repeat cycle will be calledsecond cool. Repeatability tests were also performed on

each sample type and with each scan procedure.

Representative results from heat-first DSC tests are shown in Figure 3.15. The shape

of the first-heat endotherms were ill-behaved and not repeatable, while the analogous second-

heat endotherms were repeatable. Similar results with regard to the first-heat and second-heat

endotherms are evident in the representative cool-first signatures shown in Figure 3.16. These

51

Temperature, oC

Uns

ubtr

acte

dH

eatF

low

,mW

0 50 100 15017

18

19

20

21

22

23

24

First CoolSecond Cool

Figure 3.16 Cool-first DSC signatures for first- and second-cool of as-received material.

effects are attributable to first-heat recovery of the mechanical deformation induced by shearing

the small samples needed for DSC measurements from longer lengths.

Note that both cycles of the heat-first test show two distinct exothermic peaks during the cool-

down portion of the cycles; one peaking at approximately 39�C and attributable to theR-phase

transformation, the other peaking at approximately 9�C due to the usual twinned martensitic

transformation. The R-phase is a by-product of a partially annealed condition and does not

adversely affect performance, as observed by Otsuka and Wayman [34]. Note that the cool-first

52

signature does not show the martensitic transformation on the first-cool portion of the cycle,

indicating that the alloy was delivered in a fully transformed state. From these DSC tests it can

be seen that the transformation temperatures for the Nitinol in the as-received condition and in a

free configuration are approximatelyAs=45�C (113�F), Af=60�C (140�F),Ms=17�C (62.6�F),

andMf=0�C (32�F). Note that the transformation start and finish temperatures indicated here

mark the onset/termination of substantial phase transformation activity. Measurable activity can

be seen in the tails of the exothermic and endothermic peaks of the DSC signatures. This

observation will be revisited later in this section. The DSC results give significant insight into

the state of the alloy, but mechanical properties are needed to model the behavior of the material.

Data pertaining to the material property and performance characteristics of the alloy were

acquired through mechanical testing performed at Lockheed Martin Astronautics (LMA) [43]

and at NASA LaRC. Although rigorous training could have been performed on the alloy to

generate a thermomechanically stable actuator material, the elevated cure temperature of the

glass-epoxy matrix material (176.7�C, 350�F) was considered to be high enough to risk degrading

the expensive training. Thus, thermomechanical testing was performed to quantify the material

characteristics as a function of thermal cycle. Discussion will focus on testing performed at

Lockheed Martin first, followed by the testing performed at NASA LaRC.

Isolength (constant strain) and isothermal tests were performed on nominal 0.254 m (10

inch) lengths of the material. Isolength tests were performed to measure the maximum recovery

stress generated by the actuator material as a function of thermal cycle and initial strain. The

results of this test are shown in Figure 3.17 for prestrain values of 2%, 3% and 4%. It can be

seen that in all cases the recovery stress diminishes rapidly out to about 10 thermal cycles and

approaches an asymptotic value at 50 cycles. Additional isolength tests were performed with

the 2% and 4% prestrain samples after the thermal cycling tests to measure the variation of the

stabilized recovery stress with temperature. These results are presented in Figure 3.18. Note

53

Thermal Cycle Number

Max

Rec

ove

ryS

tres

s,M

Pa

0 10 20 30 40 50300

350

400

450

500

550

600

650

700

2% Prestrain3% Prestrain4% Prestrain

Figure 3.17 Max recovery stress versus thermal cycle numberfor Nitinol ribbon samples with 2%, 3%, and 4% prestrain.

54

Temperature, oC

Rec

over

yS

tres

s,M

Pa

20 30 40 50 60 70 80 90 1000

50

100

150

200

250

300

350

400

450

500

2% Prestrain4% Prestrain

Figure 3.18 Recovery stress versus temperature for Nitinolribbon samples with 2% and 4% prestrain, after 50 cycles.

that the recovery force starts to increase at a temperature very near ambient (�24�C, 75�F). This

phenomenon appears to be in disagreement with the DSC results, but closer inspection reveals

that it is not. The initial increase in the recovery force is associated with the transformation

activity in the low-temperature “tail”, below the so-calledAs, of the exothermic peak in Figures

3.15 and 3.16. The abrupt change in the slope of the recovery stress vs. temperature curve,

near 40�C (104�F), is associated withAs. Also note that the higher prestrain level generates

55

greater recovery stress as expected. A couple of more subtle points can also be seen in this plot.

Although the increased recovery stress at 4% prestrain does inhibit completion of the reverse

transformation more than at 2%, as evidenced by the higher temperature as which it approaches

an asymptotic value, the maximum slope of the 2% recovery stress vs. temperature curve is

not as great as that for 4%. This trend is interesting in contrast to the data presented by Cross

et al. [28], where reduction in the peak values of the recovery stress at lower prestrain levels

is accompanied by higher maximum slope of the recovery stress vs. temperature curve. These

insights will be revisited later in this chapter as well as in Chapter 8.

A couple of options exist, with the available data from LMA, for modeling the behavior of

the SMA material in the hybrid composite for comparison to experiment. The specimens could

be thermally cycled enough (e.g., 50 times) to render the structure’s properties stable and simply

model the SMA recovery stress behavior according to Figure 3.18. Alternatively, an estimate

of the variation of recovery stress with thermal cycle number and temperature can be calculated

from the available data. Construction of such a data set can be accomplished by normalizing the

recovery stress versus temperature data to the corresponding maximum recovery stress versus

cycle data set. The resulting carpet plots are shown in Figures 3.19 and 3.20 for the samples

with 2% and 4% prestrain, respectively. Usage of these data sets allows modeling of the material

system without any requirement on thermal cycling of the SMAHC specimens.

Isothermal tests were conducted to measure load versus deflection on all of the previously

described specimens, as well as a specimen in the as-received condition, at ambient and elevated

temperature (120�C) to obtain estimates of the Young’s modulus. Representative plots, converted

to units of stress versus strain, are shown for the cases of 0% (as-received) and 4% prestrain

at ambient and elevated temperatures in Figures 3.21 and 3.22. The martensite variant

re-orientation yield stress and initiation of linear elastic behavior of the martensitic phase are

evident in the ambient temperature curves. Note the contrast between the as-received and 4%

56

0

100

200

300

400

500

600

700

Rec

ove

ryS

t re s

s ,M

Pa

0 10 20 30 40 50Thermal Cycle 20 30 40 50 60 70 80 90 100

Temperature,o C

Figure 3.19 Recovery stress versus thermal cycle and temperature for sample with 2% prestrain.

57

0

100

200

300

400

500

600

700

Rec

ove

ryS

t re s

s ,M

Pa

0 10 20 30 40 50Thermal Cycle 20 30 40 50 60 70 80 90 100

Temperature,o C

Figure 3.20 Recovery stress versus thermal cycle temperature for sample with 4% prestrain.

58

Strain

Str

ess,

MP

a

0 0.01 0.02 0.03 0.04 0.050

50

100

150

200

250

300

350

400

Ambient120oC

Figure 3.21 Applied stress versus strain for 0.09x0.006x10 inch samplewith 0% prestrain (as-received) at ambient temperature and 120�C.

59

Strain

Str

ess,

MP

a

0 0.01 0.02 0.03 0.04 0.050

50

100

150

200

250

300

350

400

Ambient120oC

Figure 3.22 Applied stress versus strain for 0.09x0.006x10 inchsample with 4% prestrain at ambient temperature and 120�C.

60

Table 3.7 Summary of Nitinol characterization tests.

Sample Prestrain, % # of CyclesMaximum Stress, MPa

Initial FinalYoung’s Modulus, GPa

120�C Ambient

1 1.99 50 429.4 296.0 55.50 22.75

2 3.08 50 533.0 363.3 62.47 25.10

3 3.07 10 517.1 396.3 56.06 24.41

4 4.00 50 586.3 412.7 55.64 24.34

5 3.95 10 579.2 442.1 55.92 24.20

6 N/A 0 N/A N/A 59.78 14.41

ambient temperature results. The 4% curve does not exhibit the long martensite variant re-

orientation plateau because the prior 4% prestrain has diminished the available variants for

re-orientation. However, the curves associated with the unique austenitic phase are linear and

practically identical, as expected.

Ambient temperature Young’s modulus estimates were calculated from only the initial

(0–25.5 MPa, 0–3.7 ksi) portion of the ambient temperature curve, while the overall slope

of the elevated temperature data was used for hot Young’s modulus estimates. Recall that the

maximum normal strain is not expected to exceed 500�� in the configurations considered in

this study. Therefore, evaluation of the ambient temperature modulus over the small range is

considered appropriate. The resulting modulus data and values summarizing the isolength tests

are shown in Table 3.7.

Recall that in the SMAHC specimen fabrication procedures, described earlier in this chapter,

the Nitinol ribbon was prestrained 4%. Although data was collected at LMA for this prestrain

level, there is a fundamental inconsistency. The ribbon used in the fabrication procedures was

necessarily released from tension after prestraining and before embedding in the composite,

whereas the recovery force tests at LMA were performed immediately after prestraining and

without regripping. Also, there was some suspicion that aging, after prestraining, may have

some effect on the actuator performance. Therefore, additional isolength and isothermal tests

61

Spacer

Power Cables

Alignment/Grip Apparatusw/ Insulators

Nitinol Ribbon

Figure 3.23 Apparatus for installation of a Nitinol ribbon samplein a tensile test machine for recovery force measurement.

were performed on excess ribbon with 4% prestrain remaining from the fabrication process.

Isolength tests were performed on 0.127 m (5 inch) gage lengths to measure the recovery

force versus temperature and thermal cycle. An adaptation of the prestraining apparatus was

used to configure the Nitinol ribbon for measurement of recovery force. The apparatus, shown

in Figure 3.23, allowed for convenient and repeatable installation of ribbon samples in the test

machine. A picture of a sample installed in the test machine, after spacer removal, is shown in

Figure 3.24. The ribbon was subjected to a slight preload�4.45 N (1 lbf) to keep the sample

taut prior to fixing the test machine grips (stroke control) and exposing the sample to thermal

cycling. The ribbon was electrically heated to expedite the thermal cycling process. The recovery

force and ribbon temperature at three locations were measured at 2–second intervals during each

thermal cycle, each of which was completed in approximately 15 minutes. The raw data was then

processed to estimate the recovery stress with a common temperature resolution and averaged

over three samples. The resulting recovery stress versus temperature data corresponding to the

second through fourth thermal cycle are shown in Figure 3.25. The data for the first thermal

62

MTS HydraulicGrip

Nitinol Alignment/Grip Apparatus

Nitinol RibbonThermocouples

Figure 3.24 Recovery force measurement in progress.

63

Temperature, oC

Rec

over

yS

tress

,MP

a

50 100 1500

100

200

300

400

500

600

LaRC, Thermal Cycle 2LaRC, Thermal Cycle 3LaRC, Thermal Cycle 4LaRC, Thermal Cycle 50

Figure 3.25 Recovery stress versus temperature of the Nitinol ribbon(prestrained 4% and released) for thermal cycles 2–4 and 50.

cycle are not included as that information is applicable only to the SMAHC cure cycle. The

fiftieth cycle data is included for comparison with the LMA data.

It can be seen by comparing the maximum recovery force with the data in Figure 3.17 that

the performance of the Nitinol ribbon that was prestrained 4% and released is similar to that

of the 3% prestrain data collected at LMA. Analogous curves were generated from the LMA

data by averaging 2% and 4% prestrain data sets in Figure 3.18 and scaling the resulting load

64

Temperature, oC

Rec

over

yS

tress

,MP

a

50 100 1500

100

200

300

400

500

600

LMA, Thermal Cycle 2LMA, Thermal Cycle 3LMA, Thermal Cycle 4LMA, Thermal Cycle 50

Figure 3.26 Recovery stress versus temperature of the Nitinol ribbon for 3%prestrain constructed from the LMA data for thermal cycles 2–4 and 50.

vs. temperature data by the maximum load for 3% prestrain at cycle numbers 2, 3, 4 and 50

in Figure 3.17. The resulting curves are shown in Figure 3.26. Note that the 3% prestrain

case “constructed” from the LMA data achieves roughly the same maximum recovery force, but

over a much more narrow temperature range. This trend is a real physical phenomenon and is

apparently attributable to aging effects. Also note the shape change with thermal cycle of the

recovery stress vs. temperature curve, evident in Figure 3.25. This effect is not reflected in

65

the data set constructed from the LMA data and expresses the importance of quantifying the

recovery stress functionality with temperature vs. thermal cycle.

Isothermal tests were conducted using 0.127 m (5 inch) gage lengths with the same gripping

apparatus as shown previously. The entire assembly, including high-temperature MTS grips,

was contained within a box furnace. Modulus tests were conducted on 3 ribbon samples at

each of 6 temperatures; 21.11�C (70�F), 37.78�C (100�F), 65.56�C (150�F), 93.33�C (200�F),

121.1�C (250�F), and 148.9�C (300�F). Estimates from the three samples were averaged and the

resulting modulus versus temperature behavior is shown in Figure 3.27. Stress-strain behavior

similar to that shown in Figure 3.22 was observed, but the modulus estimates are somewhat

different. These differences are also probably attributable to aging effects, akin to the rubber-

like pseudoelasticity behavior.

Note that the temperature range over which the modulus exhibits the most variation in

Figure 3.27 is more narrow than theAs–Af range interpreted from Figure 3.25. This obervation

is consistent with the data collected by Cross et al. [28]. The low-temperature transition

point is associated with the temperature at which there is significant austenite and high-enough

temperature to not allow immediate transformation to martensite upon stress application. The

high-temperature transition point corresponds to the temperature at which the austenite volume

fraction begins to dominate.

Published data will be used for Poisson’s ratio (0.3) and the transverse direction CTE for

the martensitic 6.6e-6 /�C (3.67e-6 /�F) and austenitic 11e-6 /�C (6.11e-6 /�F) as measure-

ment of these properties as a function of temperature would require significant additional ef-

fort/development. In light of the previous discussions concerning the differences between the

data collected at LaRC and LMA, the data collected at LaRC are more representative of the

thermomechanical properties of the Nitinol ribbon actually embedded in the composite and is

more accurate. Therefore, properties interpreted from this data were used in making the predic-

66

Temperature, oC

Mod

ulus

,GP

a

50 100 15010

20

30

40

50

60

70

80

Figure 3.27 Modulus for the Nitinol ribbon prestrained4% and released as a function of temperature.

tions shown later in this document. These properties are tabulated in Table 3.8. Note that the

modulus data has been linearly interpolated within the more coarse temperature resolution. The

2-direction CTE has been interpolated between two transition points noted in regard to Figure

3.27, 43.33�C (110�F) and 82.22�C (180�F), to approximate the variation within the reverse

transformation temperature range. The “in-service” austenite start and finish temperatures, as

interpreted from the recovery stress versus temperature data, are 26.7�C (80�F) and 148.9�C

67

(300�F), respectively. However, the temperature range of property variation is better modeled

by the temperatures at which the recovery stress vs. temperature curve rapidly changes slope,

as discussed previously.

Table 3.8 Summary of the thermomechanical properties of the Nitinol ribbon.

Recovery Stress

Temperature Cycle 2 Cycle 3 Cycle 4 E �2�C MPa MPa MPa GPa ��/�C

21.1 0 0 0 27.17 6.606

26.7 2.130 1.276 2.130 24.82 6.606

32.2 7.233 5.530 7.233 22.41 6.606

37.8 16.18 13.20 14.05 20.06 6.606

43.3 31.92 28.95 28.52 25.72 6.606

48.9 70.65 70.23 78.74 31.37 7.236

54.4 135.4 137.9 151.9 36.96 7.866

60.0 192.0 199.6 217.9 42.61 8.496

65.6 240.5 256.6 269.4 48.27 9.108

71.1 297.1 301.8 309.8 54.88 9.738

76.7 341.8 341.8 341.3 61.43 10.37

82.2 374.5 373.7 372.0 64.19 11.00

87.8 402.2 399.2 399.2 63.16 11.00

93.3 425.6 419.2 418.8 62.06 11.00

98.9 449.9 438.8 437.5 63.92 11.00

104.4 469.0 451.6 448.2 65.78 11.00

110.0 481.0 464.8 458.8 67.64 11.00

115.6 491.6 475.8 467.3 69.50 11.00

121.1 500.1 485.6 472.9 71.36 11.00

126.7 508.6 490.3 478.8 70.81 11.00

132.2 515.0 496.7 481.8 70.33 11.00

137.8 520.5 500.5 485.2 69.78 11.00

143.3 523.5 502.2 487.3 69.29 11.00

148.9 527.8 503.9 490.7 68.74 11.00

68

In this chapter, the glass-epoxy and Nitinol ribbon material system selected for this study

was described. Discussion of the procedures for fabricating the SMAHC beam specimens was

followed by results from thermomechanical characterization tests on the constituent materials. A

full set of temperature dependent material properties were generated to quantify the constitutive

model developed in the previous chapter. This constitutive model can now be incorporated in

a thermomechanical model to predict the behavior of structures fabricated from this material

system. One such model is developed using a finite element approach in the following chapter.

69

Chapter 4Finite Element Formulation

Previous models have been limited in their ability to predict the behavior of SMAHC

structures because of inaccurate/impractical constitutive models and/or classical formulation

approach. The present formlulation incorporates the ECTEM and will be developed using a finite

element approach. This combination will later be shown to accurately capture the mechanics of

the system and to afford the generality/versatility to allow comparison with experiment.

The governing equations for the geometrically nonlinear, static thermal response and the

geometrically linear dynamic response of a rectangular panel-type structure subjected to com-

bined thermal and mechanical loads is derived in this chapter. It is necessary to account for

geometrically nonlinear out-of-plane deflection because of the potential for large static deflec-

tions due to thermal buckling. Therefore, a finite element with cubic out-of-plane interpolation

functions is deemed necessary.

4.1 Element Displacement Functions

A rectangular 24 degree-of-freedom (dof) plate element [44] was chosen for the subsequent

finite element formulation. The element has four nodes with two in-plane and four out-of-plane

dof per node, and is shown schematically in Figure 4.1. The in-plane displacement functions

are bilinear and that for the out-of-plane displacement is bi-cubic:

u(x; y) = a1 + a2x+ a3y + a4xy

v(x; y) = a5 + a6x+ a7y + a8xy

(4.1)

70

x,u

y,v

z,w

1 2

34

b

a

Figure 4.1 Schematic of 24 dof rectangular plate element.

and

w(x; y) = c1 + c2x+ c3y + c4x2 + c5xy + c6y

2 + c7x3 + c8x

2y + c9xy2 + c10y

3

+ c11x3y + c12x

2y2 + c13xy3 + c14x

3y2 + c15x2y3 + c16x

3y3(4.2)

where theai and ci are generalized coordinates.

The displacement functions can also be written in a form particularly suited to numerical

implementation in terms of the nodal degrees of freedom and appropriate bilinear and bi-cubic

interpolation functions of the element coordinates :

u(x; y) =4X

i=1

Li(�; �)ui =1

ab[(a� x)(b� y)u1 + x(b� y)u2 + xyu3 + (a� x)yu4]

v(x; y) =

4X

i=1

Li(�; �)vi =1

ab[(a� x)(b� y)v1 + x(b� y)v2 + xyv3 + (a� x)yv4]

(4.3)

71

and

w(x; y) =20X

i=4

Li(x; y) qi

=1

a3b3

��a3 + 2x3 � 3ax2

��b3 + 2y3 � 3by2

�w1 +

�3ax2 � 2x3

��b3 + 2y3 � 3by2

�w2

+�3ax2 � 2x3

��3by2 � 2y3

�w3 +

�a3 + 2x3 � 3ax2

��3by2 � 2y3

�w4

+ ax(x� a)2�b3 + 2y3 � 3by2

��@w@x

�1

+ a�x3 � ax2

��b3 + 2y3 � 3by2

��@w@x

�2

+ a�x3 � ax2

��3by2 � 2y3

��@w@x

�3

+ ax(x� a)2�3by2 � 2y3

��@w@x

�4

+ b�a3 + 2x3 � 3ax2

�y(y � b)2

�@w@y

�1

+ b�3ax2 � 2x3

�y(y � b)2

�@w@y

�2

+ b�3ax2 � 2x3

��y3 � by2

��@w@y

�3

+ b�a3 + 2x3 � 3ax2

��y3 � by2

��@w@y

�4

+ abxy(x� a)2(y � b)2� @2w

@x@y

�1

+ abxy�x2 � ax

�(y � b)2

� @2w

@x@y

�2

+ abxy�x2 � ax

��y2 � by

�� @2w

@x@y

�3

+ abxy(x� a)2�y2 � by

�� @2w

@x@y

�4

�

(4.4)

whereui andvi are the in-plane dof,qi denotes the out-of-plane dof (w1 !�

@2w@x@y

�4

), anda/b

are the element length/width, respectively. It will be advantageous for discussions that follow

to cast these equations in matrix form8<:u

v

w

9=; =

24bLuc 0bLvc 00 bLwc

35�amab

�= [Luvw]fambg (4.5)

where the nodal degrees of freedom have been collected into membranefamg and bendingfabg

components

bamc = bu1 u2 u3 u4 v1 v2 v3 v4 c

babc = bw1 . . . w4 w;x1 . . . w;x4 w;y1 . . . w;y4 w;xy1 . . . w;xy4 c(4.6)

Also, the displacement shape functions have been collected into vectors

bLuc = bL1 L2 L3 L4 0 0 0 0 c

bLvc = b0 0 0 0 L1 L2 L3 L4 c

bLwc = bL5 L6 . . . L20 c

(4.7)

72

Recall the von K´arman strain displacement relations, equation (2.32). The membrane

midplane strains and midplane curvatures can be written in terms of the nodal degrees of freedom

and interpolation functions by substituting from equations (4.5) and (4.7):

f�omg =

8<:

u;xv;y

u;y+v;x

9=; =

8<:

bLu;xcbLv;yc

bLu;yc+ bLv;xc

9=;famg = [Cm]famg

f�g = �

8<:w;xxw;yy2w;xy

9=; = �

8<:

bLw;xxcbLw;yyc2bLw;xyc

9=;fabg = [Cb]fabg

(4.8)

where

[Cm] =

24L1;x L2;x L3;x L4;x 0 0 0 0

0 0 0 0 L1;y L2;y L3;y L4;yL1;y L2;y L3;y L4;y L1;x L2;x L3;x L4;x

35 (4.9)

and

[Cb] = �

24 L5;xx L6;xx . . . L20;xxL5;yy L6;yy . . . L20;yy2L5;xy 2L6;xy . . . L20;xy

35 (4.10)

The bending midplane strain requires some manipulation to obtain a useful form. One approach

follows from recognition that the second-order (nonlinear) terms can be expressed as the product

of first-order terms:

f�obg =1

2

8<:

w;2xw;2y

2w;xw;y

9=; =

1

2

24w;x 0

0 w;yw;y w;x

35�w;xw;y

�(4.11)

It will be useful later to make the definition

[�] =

24w;x 0

0 w;yw;y w;x

35

fGg =

�w;xw;y

�(4.12)

The vectorfGg can be written in terms of nodal degrees of freedom as before in equations

(4.8) to give

fGg =

�bLw;xcbLw;yc

�fabg = [C�]fabg (4.13)

73

where

[C�] =

�L5;x L6;x . . . L20;xL5;y L6;y . . . L20;y

�(4.14)

Thus, the (nonlinear) bending midplane strain can be written as follows

f�obg =1

2[�][C�]fabg (4.15)

where [�] is a constant matrix whose entries are evaluated based upon the nodal degrees of

freedom from the previous iteration in an iterative procedure. Such a procedure will be described

in Chapter 5.

[�]i+1 =

24bLw;xc 0

0 bLw;ycbLw;yc bLw;xc

35fabgi (4.16)

The form for the von K´arman midplane strain and curvature can be summarized as follows.

f�g = f�og+ zf�g

= f�omg+ f�obg+ zf�g

= [Cm]famg+1

2[�][C�]fabg+ z[Cb]fabg

(4.17)

Finally, it will be useful in subsequent discussions to write the resultant force components from

equation (2.38) in terms of the nodal degrees of freedom:

fNmg = [A][Cm]famg

fN�g =1

2[A][�][C�]fabg

fNbg = [B][Cb]fabg

(4.18)

4.2 Euler-Lagrange Equations

The equations of motion for a mechanical system can be derived following a Newtonian

approach or a variational approach. Variational methods have proven to be very effective tools

in deriving the equations of motion for complex mechanical systems. The governing equations

for the mechanical system considered in this study will be derived using the extended Hamilton’s

74

principle, as described in Meirovitch [45] for example. Hamilton’s principle states that the path

satisfying Newton’s law extremizes the time integral of the Lagrangian with respect to all possible

paths between the limits of the time interval defining prescribed initial and final configurations.

The extended Hamilton’s principle simply extends this idea to systems with non-conservative

forces:

�

t2Z

t1

(T � V +W )dt = 0 (4.19)

and

T =

Z

D

1

2�f _usg

T f _usgdD

V =

Z

D

�dD

W =

Z

D

fbgT fus � uggdD +

Z

S

ftgTfus � uggdS

(4.20)

where� is the variational operator,T is the kinetic energy,V is the elastic potential (strain)

energy,W is the work done by non-conservative forces,� is the mass density of the structure,�

is the strain energy density,fbg is the body force vector,ftg is the surface traction vector,fusg

is the displacement vector of the structure,fugg is the displacement vector of the grip/boundary

(assumed uniform over the boundary to allow for base excitation of the structure through a

fixture), D indicates the domain (volume) of the structure, andS indicates the surface of the

structure. The displacement of the structure relative to the boundary can be expressed as

fug = fus � ugg (4.21)

such that the energy terms can be rewritten in terms of the relative displacements:

T =

Z

D

1

2�f _u+ _ugg

T f _u+ _uggdD

V =

Z

D

�dD

W =

Z

D

fbgT fugdD +

Z

S

ftgTfugdS

(4.22)

75

Consideration is restricted to a panel-type structure subjected to a steady-state thermal

excitation and mechanical excitation consisting of surface pressure, out-of-plane concentrated

forces, and/or inertial loading. The following assumptions are made for the development of the

governing equations:

1. Body forces other than inertia are negligible.

2. The structure is in a state of plane stress.

3. von Karman deformation relations are valid.

4. Linear elastic material properties are valid.

Assumptions 1 and 4, and the above-mentioned restrictions allow the potential energy and non-

conservative work terms to be simplified as follows.

V =

Z

D

1

2f�gT f�gMdD =

Z

D

1

2f�gTMf�gdD

W =

Z

S

�pw +

NfXi=1

f 0

iw �(fxg � fxgi)�dS

(4.23)

wheref�g and f�gM are the stress and mechanical strain vectors, respectively,w is the out-

of-plane displacement of the structure,p is the surface pressure,f 0

i are concentrated forces

applied normal to the plane of the structure,Nf is the number of concentrated forces,fxg is

the coordinate vector in the plane of the structure,fxgi designates the point(s) of concentrated

force application, and�(fxg � fxgi) is a spatial Dirac delta function defined by

�(fxg � fxgi) = 0 for fxg 6= fxgi (4.24)

Assumptions 2 and 3 allow the constitutive relations for a single lamina to be written as shown

in the previous chapter, equation (2.26):

f�gk =��Q�k(f�gk � f�gTk)

=��Q�k(f�0g+ zf�g �

TZ

T0

f�(� )gkdT )(4.25)

76

Recall that the mechanical strain is equal to the difference between the total strain and the

thermal strain (f�gM = f�g � f�gT ). Substituting this expression into the form for the strain

energy in equation (4.23) gives

V =

Z

S

h

2Z

�

h

2

1

2

�f�g

Tk � f�g

TTk

��Q�k(f�gk � f�gTk)dD

=

Z

S

h

2Z

�

h

2

1

2

�f�gTk

��Q�kf�gk � 2f�gTk

��Q�kf�gTk � f�gTTk

��Q�kf�gTk

�dD

(4.26)

Commuting the variation and integration operations in equation (4.19) results in the following

expression for Hamilton’s principle.

t2Z

t1

(�T � �V + �W )dt = 0 (4.27)

Consider the first variation of the kinetic energy, from equations (4.22):

�T = �

Z

D

1

2�f _u+ _ugg

T f _u+ _uggdD (4.28)

Commuting the variation and integration operations again and integrating through the thickness

results in the expression

�T =

Z

D

1

2��f� _ugT f _u+ _ugg+ f _u+ _ugg

T f� _ug�dD

=

Z

D

�f� _ugT f _u+ _uggdD

=

Z

S

�hf� _ugT f _u+ _uggdS

(4.29)

where� _ug has been omitted because the instantaneous constraints are assumed to be held constant

during application of a virtual displacement to achieve the varied path. Similarly, the first

77

variation of the strain energy in equation (4.26) can be written as

�V = �

Z

S

h

2Z

�

h

2

1

2

�f�gT

k

��Q�kf�g

k� 2f�gT

k

��Q�kf�g

Tk� f�gT

Tk

��Q�kf�g

Tk

�dD

=

Z

S

h

2Z

�

h

2

1

2

�f��gT

k

��Q�kf�g

k+ f�gT

k

��Q�kf��g

k� 2f��gT

k

��Q�kf�g

Tk

�dD

=

Z

S

h

2Z

�

h

2

�f��g

T

k

��Q�kf�g

k� f��g

T

k

��Q�kf�g

Tk

�dD

(4.30)

Incorporating the von K´arman strain-displacement relations, equations (2.32), gives

�V =

Z

S

h

2Z

�

h

2

h(f��0gT+zf��gT )

��Q�k

�f�0g+ zf�g

��(f��0gT+zf��gT )

��Q�k

TZ

T0

f�(� )gkd�idD

(4.31)

Expanding the integrand results in

�V =

Z

S

h

2Z

�

h

2

hf��0gT

��Q�kf�0g+ zf��0gT

��Q�kf�g+ zf��gT

��Q�kf��0g+ z2f��gT

��Q�kf��g

� f��0gT��Q�k

TZ

T0

f�(� )gkd� � zf��gT

��Q�k

TZ

T0

f�(� )gkd�idD

(4.32)

and integrating through the thickness generates the stress resultant notation from equations (2.35).

The following form for the first variation of the strain energy results.

�V =

Z

S

�f��0gT [A]f�0g+ f��0gT [B]f�g+ f��gT [B]f�0g+ f��gT [D]f�g

� f��0gT fN�T g � f��gT fM�T g�dS

(4.33)

Finally, referring to equation (4.23) the first variation of the non-conservative work has the form

�W = �

Z

S

�pw +

NfXi=1

f 0iw �(fxg � fxg

i)�dS

=

Z

S

�p�w +

NfXi=1

f 0i�w �(fxg � fxg

i)�dS

(4.34)

78

Recall that�(fxg � fxgi) is a spatial Dirac delta function and shouldn’t be confused with the

variational operator.

The form of Hamilton’s principle given in equation (4.27) can now be rewritten by substi-

tuting from equations (4.29), (4.33), and (4.34):t2Z

t1

hZ

S

��hf� _ugT f _u+ _ugg � f��0gT [A]f�0g � f��0gT [B]f�g � f��gT [B]f�0g � f��gT [D]f�g

+ f��0gTfN�T g+ f��gTfM�T g+ p�w +

NfXi=1

f 0

i�w �(fxg � fxgi)dSidt = 0

(4.35)

The first term can be integrated by parts to giveZ

S

�hf�ugTt2

jt1

f _u+ _uggdS �

t2Z

t1

hZ

S

��hf�ugT f�u+ �ugg � f��0gT [A]f�0g � f��0gT [B]f�g

� f��gT [B]f�0g � f��gT [D]f�g+ f��0gT fN�Tg

+ f��gTfM�T g+ p�w +

NfXi=1

f 0

i�w �(fxg � fxgi)dSidt = 0

(4.36)

Note that the first term in the latter equation is equal to zero because the integrand vanishes as

the virtual displacementsf�ug are defined to be zero at timest1 and t2.

The expression given in equation (4.36) resulted from application of extended Hamilton’s

principle to the energy terms associated with the entire structure. The same approach is applicable

to any portion of the structure, e.g., finite element, due to physical constraints. Therefore, the

following development will focus on the element level. Recall that expressions for the midplane

membrane strains��0

and curvaturesf�g were given in terms of element degrees of freedom in

equations (4.17). The variation of the midplane strain within the element would then be written as

f��0g = [Cm]f�amg+1

2[��][C�]fabg+

1

2[�][C�]f�abg (4.37)

or

f��0g = [Cm]f�amg+1

2[��]fGg+

1

2[�]f�Gg (4.38)

79

by substituting from equation (4.13). The components of the second and third terms can be

examined as follows

1

2[��][C�]fabg+

1

2[�][C�]f�abg =

1

2

0@�24w;x 0

0 w;yw;y w;x

351A�w;xw;y

�+

1

2

0@24w;x 0

0 w;yw;y w;x

351A�

�w;xw;y

�

=1

2

8<:

�w;xw;x�w;yw;y

�w;y w;x+�w;xw;y

9=; +

1

2

8<:

w;x �w;xw;y �w;y

w;y �w;x+w;x �w;y

9=;

(4.39)

It can be seen that1

2[��][C�]fabg+

1

2[�][C�]f�abg =

1

2[�]f�Gg+

1

2[�]f�Gg

= [�]f�Gg

= [�][C�]f�abg

(4.40)

So, the final forms for the membrane strain and curvature vectors in terms of element degrees

of freedom are as follows.

f��0g = [Cm]f�amg+ [�][C�]f�abg

f��g = [Cb]f�abg

(4.41)

The structural displacement-related quantities in equation (4.36) can be written in terms of the

element degrees of freedom from equation (4.5):

f�ugT f�u+ �ugg = f�ambgT [Luvw]

T [Luvw]f�amb + �ugg

=

��am�ab

�T �fLug fLvg 0

0 0 fLwg

�24bLuc 0

bLvc 0

0 bLwc

35��am�ab

+0

�ug

�

=

��am�ab

�T �fLugfLug

T + fLvgfLvgT 0

0 fLwgfLwgT

��am�ab

+0

�ug

�

= f�amgT�fLugfLug

T+ fLvgfLvgT�f�amg+f�abg

T fLwgfLwgT f�ab+�ugg

(4.42)

and

�w = fLwgTf�abg = f�abg

T fLwg (4.43)

where it has been assumed that the base acceleration component�ug is normal to the plane of

the structure. The expressions in equations (4.41), (4.42), and (4.43) can be substituted into

80

Hamilton’s principle, equation (4.36), to give

t2Z

t1

h ZAe

��h(f�amg

T�fLugfLug

T + fLvgfLvgT�f�amg+ f�abg

T fLwgfLwgT f�ab + �ugg)

+ ([Cm]f�amg+ [�][C�]f�abg)T [A]([Cm]famg+

1

2[�][C�]fabg)

+ ([Cm]f�amg+ [�][C�]f�abg)T [B][Cb]fabg+ f�abg

T [Cb]T [B]([Cm]famg+

1

2[�][C�]fabg)

+ f�abgT[Cb]

T[D][Cb]fabg � ([Cm]f�amg+ [�][C�]f�abg)

TfN�T g � f�abg

T[Cb]

TfM�T g

� p�w �

nfXi=1

f 0

i�w �(fxg � fxgi)dAe

idt = 0

(4.44)

where the spatial integration has been limited to the areaAe of the finite element in concern and

the number of concentrated forces in the summation has been limited to that numbernf within

the element area. Expanding the integrand results in the following expression

t2Zt1

hZAe

��hhf�amg

T�fLugfLug


T fLwgfLwgT f�ab + �ugg

i

+ f�amgT [Cm]T [A][Cm]famg+

1

2f�amg

T [Cm]T [A][�][C�]fabg

+ f�abgT [C�]

T [�]T [A][Cm]famg+1

2f�abg

T [C�]T [�]T [A][�][C�]fabg

+ f�amgT [Cm]T [B][Cb]fabg+ f�abg

T [C�]T [�]T [B][Cb]fabg+ f�abg

T [Cb]T [B][Cm]famg

+1

2f�abg

T[Cb]

T[B][�][C�]fabg+ f�abg

T[Cb]

T[D][Cb]fabg � f�amg

T[Cm]

TfN�T g

� f�abgT [C�]

T [�]T fN�T g � f�abgT [Cb]

T fM�T g

� f�abgT fLwgp� f�abg

TfLwg

nfXi=1

f 0


idt = 0

(4.45)

Consider the product[�]T fNg. Expanding the product in terms of the matrix components

results in the following identity

[�]T fNg =

�w;x 0 w;y0 w;y w;x

�8<:Nx

Ny

Nxy

9=; =

�Nx Nxy

Nxy Ny

��w;xw;y

�= [N ]fGg

= [N ][C�]fabg

(4.46)

81

This identity is useful for further manipulation of equation (4.45). Note that the terms

f�abgT [C�]

T [�]T [A][Cm]famg and f�abgT [C�]

T [�]T [B][Cb]fabg (4.47)

in equation (4.45) do not have symmetric counterparts. However, noting that from equations

(4.18) [A][Cm]famg = fNmg, the former of the two terms in equation (4.47) can be split as

follows

f�abgT [C�]

T [�]T [A][Cm]famg =1

2f�abg

T [C�]T [�]T [A][Cm]famg+

1

2f�abg

T [C�]T [�]TfNmg

(4.48)

and the identity in equation (4.46) can be employed to give

f�abgT[C�]

T[�]

T[A][Cm]famg =

1

2f�abg

T[C�]

T[�]

T[A][Cm]famg+

1

2f�abg

T[C�]

T[Nm][C�]fabg

(4.49)

Now, the first half of the term has a symmetric counterpart in equation (4.45) and the second

half is symmetric itself. Similarly, noting that[B][Cb]fabg = fNbg, the latter of the two terms

in equation (4.47) can be split as follows

f�abgT[C�]

T[�]

T[B][Cb]fabg =

1

2f�abg

T[C�]

T[�]

T[B][Cb]fabg+

1

2f�abg

T[C�]

T[�]

TfNbg

(4.50)

and the identity in equation (4.46) can be employed to give

f�abgT [C�]

T [�]T [B][Cb]fabg =1

2f�abg

T [C�]T [�]T [B][Cb]fabg+

1

2f�abg

T [C�]T [Nb][C�]fabg

(4.51)

So, both terms in equation (4.47) have been transformed into symmetric forms. Finally, the

identity in equation (4.46) can be used to rewrite another term in equation (4.45) as follows.

f�abgT [C�]

T [�]T fN�T g = f�abgT [C�]

T [N�T ][C�]fabg (4.52)

82

Substituting the expressions from equations (4.49), (4.51), and (4.52) into Hamilton’s principle

for the element, equation (4.45), results in the following expression.

t2Z

t1

hZ

Ae

��hhf�amg

T�fLugfLug


T fLwgfLwgT f�ab + �ugg

i

+ f�amgT [Cm]T [A][Cm]famg+

1

2f�amg

T [Cm]T [A][�][C�]fabg

+1

2f�abg

T[C�]

T[�]

T[A][Cm]famg+

1

2f�abg

T[C�]

T[Nm][C�]fabg

+1

2f�abg

T [C�]T [�]T [A][�][C�]fabg+ f�amg

T [Cm]T [B][Cb]fabg

+ f�abgT [Cb]

T [B][Cm]famg+1

2f�abg

T [C�]T [�]T [B][Cb]fabg

+1

2f�abg

T [C�]T [Nb][C�]fabg+

1

2f�abg

T [Cb]T [B][�][C�]fabg

+ f�abgT [Cb]

T [D][Cb]fabg � f�amgT [Cm]TfN�T g

� f�abgT[C�]

T[N�T ][C�]fabg � f�abg

T[Cb]

TfM�Tg

� f�abgT fLwgp� f�abg

TfLwg

nfXi=1

f 0


idt = 0

(4.53)

Separating terms involvingf�amgT and f�abg

T gives

t2Z

t1

hZ

Ae

�f�amg

T [�h�fLugfLug

T + fLvgfLvgT�f�amg+ [Cm]T [A][Cm]famg

+1

2[Cm]

T [A][�][C�]fabg+ [Cm]T [B][Cb]fabg � [Cm]T fN�T g]

+ f�abgT [�hfLwgfLwg

T f�ab + �ugg+1

2[C�]

T [�]T [A][Cm]famg

+1

2[C�]

T [Nm][C�]fabg+1

2[C�]

T [�]T [A][�][C�]fabg+ [Cb]T [B][Cm]famg

+1

2[C�]

T [�]T [B][Cb]fabg+1

2[C�]

T [Nb][C�]fabg+1

2[Cb]

T [B][�][C�]fabg

+ [Cb]T [D][Cb]fabg � [C�]

T [N�T ][C�]fabg � [Cb]TfM�T g

� fLwgp � fLwg

nfXi=1

f 0

i�w �(fxg � fxgi)]dAe

idt = 0

(4.54)

83

The virtual displacementsf�amg andf�abg are arbitrary and independent, so the corresponding

components of the integrand in equation (4.54) must each equal zero. The two resulting equations

can be written in matrix form:

�mb 00 mm

��ab�am

�+

�kb kbmkmb km

��

�k�T 00 0

�+

1

2

�n1b + (n1b)B n1bm

n1mb 0

�

+1

3

�n2b 00 0

�!�abam

�=

�pb(t)0

�+

�fb(t)0

�+

�pb�Tpm�T

��

�mb 00 mm

��ug0

�(4.55)

where the element matrices are as follows.

[mb] =

ZAe

�hfLwgfLwgTdAe [mm] =

ZAe

�h�fLugfLug

T + fLvgfLvgT�dAe

[kb] =

ZAe

[Cb]T [D][Cb]dAe [km] =

ZAe

[Cm]T [A][Cm]dAe

[kbm] =

ZAe

[Cb]T[B][Cm]dAe [kmb] = [kbm]

T

[k�T ] =

ZAe

[C�]T [N�T ][C�]dAe [n1b] =

ZAe

[C�]T [Nm][C�]dAe

[(n1b)B] =

ZAe

([C�]T [�]T [B][Cb] + [C�]

T [Nb][C�] + [Cb]T [B][�][C�])dAe

[n1bm] =

ZAe

[C�]T[�]

T[A][Cm]dAe [n1mb] = [n1bm]

T

[n2b] =3

2

ZAe

[C�]T [�]T [A][�][C�]dAe

fpb(t)g =

ZAe

p(x; y; t)fLwgdAe ffb(t)g =

nfXi=1

ZAe

f 0

i(t) �(fxg � fxgi)fLwgdAe

fpb�T g =

ZAe

[Cb]T fM�T gdAe fpm�T g =

ZAe

[Cb]TfN�T gdAe

(4.56)

The element area integrations can be conveniently and efficiently evaluated though Gauss-

Legendre numerical integration, as demonstrated in Appendix B. The governing equations for

84

each element can be assembled to form the Euler-Lagrange equations for the entire system:�Mb 0

0 Mm

��Ab�Am

�+

��Kb KB

KTB Km

��

�K�T 0

0 0

�

+1

2

�N1b + (N1b)B N1bm

N1mb 0

�+

1

3

�N2b 00 0

��Ab

Am

�

=

�Pb(t)0

�+

�Fb(t)0

�+

�Pb�T

Pm�T

��

�Mb 00 Mm

��Ug0

�(4.57)

or

[M ]n�Ao+�[K]� [K�T ] +

1

2[N1] +

1

3[N2]

�fAg = fP (t)g+ fF (t)g+ fP�T g � [M ]f �Ug(t)g

(4.58)

where [M ] and [K] are the usual system mass and linear stiffness matrices;[K�T ] is the

geometric stiffness matrix due to the thermal in-plane force vectorfN�Tg; [N1] and [N2] are

the first- and second-order nonlinear stiffness matrices which depend linearly and quadratically

upon displacementfAg, respectively;fP (t)g is the surface pressure excitation load vector,

fF (t)g is the concentrated force excitation load vector,fP�T g is the thermal force vector, and

f �Ug(t)g is the inertial excitation base acceleration vector. The subscriptsb andm denote bending

and membrane components, respectively, and the subscriptB indicates that the corresponding

stiffness matrix is due to the laminate bending-membrane coupling stiffness matrix [B]. Note

that the stiffness matrices[K], [K�T ], [N1], [N2] and the thermal force vectorfP�T g are all

temperature dependent.

4.2.1 Separation of Static and Dynamic Components

Recall that the system is restricted to be thermally steady-state. It is assumed that the structure

will undergo small dynamic motions about a stable static shape (flat or thermally buckled) so

that the system degrees of freedom will consist of static and dynamic components

fAg = fAgs + fAgd (4.59)

such that

fAgd << fAgs ! fAg2d � 0 (4.60)

85

Consider the effect, on the element level, of equation (4.59) on the form of[�] in equation (4.12):

[�] = [�]s+ [�]d =

24ws;x 0

0 ws;yws;y ws;x

35+

24wd;x 0

0 wd;ywd;y wd;x

35 (4.61)

It can be seen from equations (4.56) that the nonlinear stiffness matrices,[n1] and[n2], are thus

affected. For example,

[n1bm] =

Z

S

[C�]T [�]T [A][Cm]dS

=

Z

S

[C�]T �

[�]Ts + [�]

Td

�[A][Cm]dS

=

Z

S

[C�]T [�]Ts [A][Cm]dS +

Z

S

[C�]T [�]Td [A][Cm]dS

= [n1bm]s + [n1bm]d

(4.62)

Similarly,

[n1mb] = [n1mb]s + [n1mb]d (4.63)

Note that the resultant forces due to membrane and bending strain, equations (4.18), are also

affected. So, these forces can be written as

fNmg = fNmgs + fNmgd and fNbg = fNbgs + fNbgd (4.64)

or

[Nm] = [Nm]s + [Nm]d and [Nb] = [Nb]s + [Nb]d (4.65)

Therefore,

[n1b] = [n1b]s + [n1b]d and [(n1b)B] = [(n1b)B]s + [(n1b)B]d (4.66)

86

Finally, the second-order nonlinear element stiffness matrix takes the form

[n2b] =3

2

Z

S

[C�]T [�]T [A][�][C�]dS

=3

2

Z

S

[C�]T �[�]Ts + [�]Td

�[A]

�[�]s + [�]d

�[C�]dS

=3

2

Z

S

�[C�]

T [�]Ts [A][�]s[C�] + [C�]T [�]Ts [A][�]d[C�]

+ [C�]T [�]Td [A][�]s[C�] + [C�]

T [�]Td [A][�]d[C�]�dS

= [n2b]s + [n2b]sd + [n2b]ds + [n2b]d

(4.67)

This distributive property conveys to the system matrices:

[N1] = [N1]s + [N1]d

[N2] = [N2]s + [N2]sd + [N2]ds + [N2]d

(4.68)

and the system equations of motion, equation (4.58), become

[M ]f �Agd + ([K]� [K�T ])(fAgs + fAgd) +1

2([N1]s + [N1]d)(fAgs + fAgd)

+1

3([N2]s + [N2]sd + [N2]ds + [N2]d)(fAgs + fAgd)

= fP (t)g+ fF (t)g+ fP�T g � [M ]f �Ugg

(4.69)

It is desirable to separate the equations governing the static and dynamic response. However,

there are terms which couple the static and dynamic displacements. The term[N2]d can be

neglected because it is second order with respect tofAgd. The form of other coupling terms

can be clarified as follows.

Consider the product of the first-order nonlinear dynamic stiffness and the static displacement

vector. On the element level, referring to equations (4.56), the four components of this matrix

can be manipulated as follows in discussions designated by numbers 1) through 4).

1) The product of the first-order dynamic bending stiffness and static bending nodal degrees of

freedom has the form

[n1b]dfabgs =

Z

S

[C�]T [Nm]d[C�]dSfabgs (4.70)

87

This integrand can be rewritten by noting the following

[Nm]d[C�]fabgs = [�]T

sfNmgd

= [�]T

s[A][Cm]famgd

(4.71)

so the expression given by equation (4.70) can be rewritten as

[n1d]dfabgs =

Z

S

[C�]T [�]T

s[A][Cm]dSfamgd (4.72)

The form of the integral in equation (4.72) can be immediately recognized as the product of

the first-order static coupling stiffness and the dynamic membrane displacements. Therefore, the

expression in equation (4.72) can be written as follows.

[n1b]dfabgs = [n1bm]sfamgd (4.73)

2a) The product of the first-order dynamic bending stiffness due to the[B] matrix and static

bending nodal degrees of freedom has the form

[(n1b)B]dfabgs =

Z

S

([C�]T [�]T

d[B][Cb] + [C�]

T [Nb]d[C�] + [Cb]T [B][�]

d[C�])dSfabgs (4.74)

but recall from equations (4.18) and (4.46) that the substitutions

[B][Cb]fabgs = fNbgs and [�]TdfNbgs = [Nb]s[C�]fabgd (4.75)

can be made. It follows that the expression given in equation (4.74) can be rewritten as

[(n1b)B]dfabgs =

Z

S

[C�]T [Nb]s[C�]dSfabgd +

Z

S

[C�]T [Nb]d[C�]dSfabgs

+

Z

S

[Cb]T [B][�]

d[C�]dSfabgs

(4.76)

2b) Conversely, the product of the first-order static bending stiffness due to the[B] matrix and

dynamic bending nodal degrees of freedom has the form

[(n1b)B]sfabgd =

Z

S

([C�]T [�]T

s[B][Cb] + [C�]

T [Nb]s[C�] + [Cb]T [B][�]

s[C�])dSfabgd (4.77)

88

but equations (4.18) and (4.46) allow the substitutions

[B][Cb]fabgd = fNbgd and [�]Ts fNbgd = [Nb]d[C�]fabgs (4.78)

It follows that equation (4.77) can be rewritten as

[(n1b)B]sfabgd =

Z

S

[C�]T [Nb]d[C�]dSfabgs +

Z

S

[C�]T [Nb]s[C�]dSfabgd

+

Z

S

[Cb]T [B][�]s[C�]dSfabgd

(4.79)

The first two terms of equations (4.76) and (4.79) are obviously identical, but the third term

requires further investigation. Recall from equation (4.13) that[�][C�]fabg = [�]fGg. Therefore,

term three in equation (4.76) can be rewritten in terms of out-of-plane displacement derivatives:

[�]d[C�]fabgs = [�]dfGgs =

24wd;x 0

0 wd;ywd;y wd;x

35�ws;xws;y

�=

8<:

wd;xws;xwd;y ws;y

wd;y ws;x+wd;xws;y

9=; (4.80)

It is clear that the previous expression is the same as the following.

[�]s[C�]fabgd = [�]sfGgd =

24ws;x 0

0 ws;yws;y ws;x

35�wd;xwd;y

�=

8<:

ws;xwd;xws;y wd;y

ws;y wd;x+ws;xwd;y

9=; (4.81)

Thus, the third term in equations (4.76) and (4.79) are identical, and the final result gives the

following equality.

[(n1b)B]dfabgs = [(n1b)B]sfabgd (4.82)

3) The product of the first-order dynamic coupling stiffness and the static membrane displace-

ments has the form

[n1bm]dfamgs =

ZS

[C�]T [�]Td [A][Cm]dSfamgs (4.83)

Equations (4.18) and (4.46) allow the substitutions

[A][Cm]famgs = fNmgs and [�]Td fNmgs = [Nm]s[C�]fabgd (4.84)

89

So, equation (4.83) can be rewritten as

[n1bm]dfamgs =

Z

S

[C�]T [Nm]Ts [C�]dSfabgd (4.85)

The form of the integral in equation (4.85) is easily recognized as the product of the first-order

static bending stiffness and the dynamic bending nodal degrees of freedom. Thus, the final

equality becomes

[n1bm]dfamgs = [n1b]sfabgd (4.86)

4) The product of the first-order dynamic coupling stiffness and the static bending nodal degrees

of freedom has the form

[n1mb]dfabgs =

Z

S

[Cm]T [A][�]d[C�]dSfabgs (4.87)

but recall from equation (4.13) that[�][C�]fabg = [�]fGg. So, the analogous product in equation

(4.87) can be written in terms of out-of-plane displacement derivatives:

[�]d[C�]fabgs = [�]dfGgs =

24wd;x 0

0 wd;ywd;y wd;x

35�ws;xws;y

�=

8<:

wd;xws;xwd;y ws;y

wd;y ws;x+wd;xws;y

9=; (4.88)

This expression can be manipulated into the following equivalent form

[�]s[C�]fabgd = [�]sfGgd =

24ws;x 0

0 ws;yws;y ws;x

35�wd;xwd;y

�=

8<:

ws;xwd;xws;y wd;y

ws;y wd;x+ws;xwd;y

9=; (4.89)

which provides the final needed equality.

[n1mb]dfabgs = [n1mb]sfabgd (4.90)

The four equalities given in equations (4.73), (4.82), (4.86), and (4.90) can be written in the

matrix form

�n1b;d + (n1b)B;d n1bm;d

n1mb;d 0

��ab;sam;s

�=

�n1b;s + (n1b)B;s n1bm;s

n1mb;s 0

��ab;dam;d

�(4.91)

90

from which it is easily seen that the assembled system matrices share the analogous equality:

[N1]dfAgs = [N1]sfAgd (4.92)

The cross-terms generated by the expansion of the second-order nonlinear stiffness into static

and dynamic components need manipulation as well. The product of each cross term with the

corresponding displacement vector, on the element level, can be manipulated as follows in the

discussions designated by numbers 10) through 40).

10) The product of the second-order bending stiffness cross-term[n2b]sd and static bending nodal

degrees of freedomfabgs is written as

[n2b]sdfabgs =

Z

S

[C�]T [�]Ts [A][�]d[C�]dSfabgs (4.93)

but recall from the previous manipulations designated by 2b) and 4) that

[�]ifGgj = [�]jfGgi ) [�]d[C�]fabgs = [�]s[C�]fabgd (4.94)

Thus, the following substitution can be made.

[n2b]sdfabgs = [n2b]sfabgd (4.95)

20) The product of the second-order bending stiffness cross-term[n2b]sd and dynamic bending

nodal degrees of freedomfabgd has the form

[n2b]sdfabgd =

Z

S

[C�]T [�]Ts [A][�]d[C�]dSfabgd (4.96)

but recall from numbers 1) and 2a) that

[A][�]d[C�]fabgd = fN�gd and [�]Ts fN�gd = [N�]d[C�]fabgs (4.97)

Thus, the expression can be rewritten as follows.

[n2b]sdfabgd =

Z

S

[C�]T[N�]d[C�]dSfabgs

= [n2b]dfabgs

(4.98)

91

30) Similarly, the product of the second-order bending stiffness cross-term[n2b]ds and static

bending nodal degrees of freedomfabgs has the form

[n2b]dsfabgs =

Z

S

[C�]T [�]Td [A][�]s[C�]dSfabgs (4.99)

With the following substitutions

[A][�]s[C�]fabgs = fN�gs and [�]Td fN�gs = [N�]s[C�]fabgd (4.100)

the expression can be rewritten as

[n2b]dsfabgs =

Z

S

[C�]T [N�]s[C�]dSfabgd

= [n2b]sfabgd

(4.101)

40) Finally, the product of the second-order bending stiffness cross-term[n2b]ds and dynamic

bending nodal degrees of freedomfabgd can be written as

[n2b]dsfabgd =

Z

S

[C�]T [�]Td [A][�]s[C�]dSfabgd (4.102)

but from number 10) it is known that

[�]ifGgj = [�]jfGgi ) [�]s[C�]fabgd = [�]d[C�]fabgs (4.103)

Thus, this term can be rewritten in the following form.

[n2b]dsfabgd = [n2b]dfabgs (4.104)

So, on the system level, implementation of the manipulations designated by 10) through 40) gives

[N2]sdfAgs = [N2]sfAgd

[N2]sdfAgd = [N2]dfAgs

[N2]dsfAgs = [N2]sfAgd

[N2]dsfAgd = [N2]dfAgs

(4.105)

92

Expanding the system equations of motion, equations (4.69), results in

[M ]f �Agd + [K]fAgs + [K]fAgd � [K�T ]fAgs � [K�T ]fAgd +1

2[N1]sfAgs

+1

2[N1]sfAgd +

1

2[N1]dfAgs +

1

2[N1]dfAgd +

1

3[N2]sfAgs +

1

3[N2]sfAgd

+1

3[N2]sdfAgs +

1

3[N2]sdfAgd +

1

3[N2]dsfAgs +

1

3[N2]dsfAgd +

1

3[N2]dfAgs

+1

3[N2]dfAgd = fP (t)g+ fF (t)g+ fP�T g � [M ]f �Ugg

(4.106)

and making the substitutions from equations (4.92) and (4.105) gives the following expression.

[M ]f �Agd + [K]fAgs + [K]fAgd � [K�T ]fAgs � [K�T ]fAgd +1

2[N1]sfAgs

+1

2[N1]sfAgd +

1

2[N1]sfAgd +

1

2[N1]dfAgd +

1

3[N2]sfAgs +

1

3[N2]sfAgd

+1

3[N2]sfAgd +

1

3[N2]dfAgs +

1

3[N2]sfAgd +

1

3[N2]dfAgs +

1

3[N2]dfAgs

+1

3[N2]dfAgd = fP (t)g+ fF (t)g+ fP�T g � [M ]f �Ugg

(4.107)

Separating the terms involving the static displacementsfAgs from those involving the dynamic

displacementsfAgd results in the following static and dynamic governing equations.

�[K]� [K�T ] +

1

2[N1]s +

1

3[N2]s + [N2]d

�fAgs = fP�T g

[M ]f �Agd +�[K]� [K�T ] + [N1]s +

1

2[N1]d + [N2]s

+1

3[N2]d

�fAgd = fP (t)g+ fF (t)g � [M ]f �Ugg

(4.108)

The matrix[N2]d, appearing in both equations, can be neglected because it is second order with

respect to the small dynamic displacementsfAgd. The matrix[N1]d, appearing in the dynamic

equation, can be neglected also because the product[N1]dfAgd is second-order infAgd as well.

Therefore, the final static and dynamic governing equations are as follows.

�[K]� [K�T ] +

1

2[N1]s +

1

3[N2]s

�fAgs = fP�T g Static

[M ]f �Agd +�[K]� [K�T ] + [N1]s + [N2]s

�fAgd

= fP (t)g+ fF (t)g � [M ]f �Ugg Dynamic

(4.109)

93

In the applications being considered for this technology, there is often a combination of a

flexible structures with a rigid boundary and combined thermal and mechanical loads. Thus,

there is an opportunity for thermal buckling and an analysis capable of predicting such an event

is necessary. Often it is also desirable to predict the post-buckled deflection of a structure

subsequent to such an event. It will be shown later in this document that certain combinations

of constituent materials in a SMAHC structure can exhibit a thermal buckling event and return

the post-buckling deflection to zero. In order to capture and better understand this behavior,

thermal buckling and post-buckling analyses are necessary.

Thus, three types of analyses are required to study the response of the structures considered

in this study: (1) thermal buckling analysis, (2) thermal post-buckling analysis, and (3) dynamic

analysis. Each of these analyses are discussed in the following two chapters. Acoustic radiation

analysis is presented in Chapter 7 to study the structural-acoustic response and noise transmission

characteristics of panel-type structures. A flowchart of the overall thermomechanical analysis

procedure is given in Figure 4.2.

94

Tp > To?

Read input data

Tp > Tcr?

Perform thermal buckling analysisStore bending eigenvector and/ormembrane deflections

Perform thermal post-buckling analysisStore bending and membrane deflections

Perform vibration andacoustic radiation analysis

No

No

Yes

Yes

Figure 4.2 Overall thermomechanical analysis flowchart.

95

Chapter 5Static Response

The static response considered in this study is assumed to be solely due to a thermal load.

Two analysis types are potentially needed to determine this response: (1) thermal buckling

analysis and (2) thermal post-buckling analysis. Each analysis will be discussed in this chapter.

Thermal buckling and post-buckling considerations necessitate the investigation of static stability.

Therefore, the fundamental equations governing this behavior will be derived first.

5.1 Static Stability and Nonlinear Analysis

The idea of static stability commonly arises in the context of a structure subjected to

compressive mechanical loads in an axial direction. The goal is typically to determine the

critical load for which the structure loses stability and may be deflected significantly by a

lateral load, from which it will not return upon lateral load removal (commonly referred to as

buckling). In practice, a structure in such a condition often deflects (buckles) as a result of

imperfections, without the application of lateral loads. Whether buckled by mechanical loads

and/or imperfections, it is often desirable to calculate the deflected shape of a structure subsequent

to buckling (often termed post-buckling deflected shape). Such an analysis usually involves

geometrically nonlinear analysis as the deflections are typically large.

In the case of thermoelastic response analysis, such as that considered here, a compressive

load is generated by thermal expansion of a structure in the presence of mechanical boundary

constraints. Thus, the goal is to determine the critical temperature at which the structure becomes

96

unstable (also called the buckling temperature). Analysis of the thermal post-buckling deflection

will also be of particular value.

Stability of an equilibrium configuration can be investigated through the use of the adjacent

equilibrium method [46, 47]. Application of this method entails imposing small increments to

the displacements of a system in static equilibrium and examining the resulting adjacent con-

figuration. Letf(A)g denote an equilibrium configuration for a structure with a displacement

field given byfAg. Now, let the static displacements be given an increment

fAg ! fAg+ f�Ag such that��A

2� f0g (5.1)

where thef�Ag are infinitesimal displacements. The adjacent static configurationf(A+ �A)g

can be approximated by a truncated Taylor Series expansion:

f(A+ �A)g = f(A)g+df(A)g

dfAgf�Ag+O

��A

2�

(5.2)

One could proceed to determine the derivative of the static configuration with respect to the

static displacements, but the necessary ingredients for an equivalent procedure has already been

developed in the previous chapter and can be employed here.

Recall that the equilibrium configurationf(A)g is governed by the static system in

equations (4.109):

f(A)g = ([K]� [K�T ] +1

2[N1] +

1

3[N2])fAg � fP�T g = f0g (5.3)

where the static subscripts has been omitted for simplicity. The expression for the increment in

the static displacements, equation (5.1), is analogous to the separation of the static and dynamic

components of the total displacements given in equation (4.59). Substitution offAg+f�Ag into

equation (5.3) gives the adjacent static configuration that is not necessarily in equilibrium:

f(A+ �A)g = ([K]� [K�T ] +1

2[N1(A+ �A)] +

1

3[N2(A+ �A)])(fAg+ f�Ag)� fP�T g

(5.4)

97

The manipulation of terms in the discussion following equation (4.59) allows the first- and

second-order nonlinear stiffness matrices for the static system in equation (5.4) to be written as

[N1] = [N1]0+ [N1]

�

[N2] = [N2]0+ [N2]

0�+ [N2]

�0+ [N2]

�

(5.5)

where the subscripts0 and� indicate the influence of the displacementsfAg andf�Ag, respec-

tively.

Expanding[N1] and [N2] in equation (5.4) gives the following expression for the adjacent

configuration.

f(A+ �A)g =�[K]� [K�T ] +

1

2([N1]

0+ [N1]

�)

+1

3([N2]

0+ [N2]

0�+ [N2]

�0+ [N2]

�)�(fAg+ f�Ag)� fP�T g

(5.6)

Further manipulation allows several terms in equation (5.6) to be written in forms analogous to

equations (4.92) and (4.105):

[N1]�fAg = [N1]

0f�Ag

[N2]0�fAg = [N2]

0f�Ag

[N2]0�f�Ag = [N2]

�fAg

[N2]�0fAg = [N2]

0f�Ag

[N2]�0f�Ag = [N2]

�fAg

(5.7)

Expanding equation (5.6) gives

f(A+ �A)g = [K]fAg+ [K]f�Ag � [K�T ]fAg � [K�T ]f�Ag+1

2[N1]

0fAg

+1

2[N1]

0f�Ag+

1

2[N1]

�fAg+

1

2[N1]

�f�Ag+

1

3[N2]

0fAg

+1

3[N2]

0f�Ag+

1

3[N2]

0�fAg+

1

3[N2]

0�f�Ag+

1

3[N2]

�0fAg

+1

3[N2]

�0f�Ag+

1

3[N2]

�fAg+

1

3[N2]

�f�Ag � fP�T g

(5.8)

98

and making the substitutions from equations (5.7) results in

f(A+ �A)g = [K]fAg+ [K]f�Ag � [K�T ]fAg � [K�T ]f�Ag+1

2[N1]

0fAg

+1

2[N1]

0f�Ag+

1

2[N1]

0f�Ag+

1

2[N1]

�f�Ag+

1

3[N2]

0fAg

+1

3[N2]

0f�Ag+

1

3[N2]

0f�Ag+

1

3[N2]

�fAg+

1

3[N2]

0f�Ag

+1

3[N2]

�fAg+

1

3[N2]

�fAg+

1

3[N2]

�f�Ag � fP�T g = f0g

(5.9)

Separating terms infAg and f�Ag, neglecting second- and higher-order terms inf�Ag, and

omitting the subscript0 gives

f(A+ �A)g =�[K]� [K�T ] +

1

2[N1] +

1

3[N2]

�fAg � fP�T g

+�[K]� [K�T ] + [N1] + [N2]

�f�Ag

(5.10)

In order for this adjacent static configuration to be in equilibrium we must havef(A+ �A)g =

f0g. Then, the form for theincrementalstatic system of equations becomes evident when the

relations in equations (5.2) and (5.3) are compared to that in equation (5.10):

�[K]� [K�T ] + [N1] + [N2]

�f�Ag = f0g (5.11)

as the original equilibrium configurationf(A)g is identically satisfied. It can also be seen

that the first-order derivative of the original equilibrium configuration with respect to the static

displacement vector is given by

df(A)g

dfAg= [Ktan] = [K]� [K�T ] + [N1] + [N2] (5.12)

which is called thetangentstiffness matrix.

It will be shown in the following sections that Equations (5.10)–(5.12) are useful for thermal

buckling and post-buckling analyses. The incremental static system of equations will be useful for

investigation of thermal buckling as the expression must be satisfied for an adjacent equilibrium

to exist. The tangent stiffness matrix will be useful in calculation of static deflections due to

thermal moments/bending-membrane coupling or post-buckling deflections, where a Newton-

Raphson approach will require the derivative of the stiffness coefficients. These procedures will

be described subsequently.

99

5.2 Thermal Buckling Analysis

A thermal buckling analysis is carried out to determine the critical (buckling) temperature

distribution of a structureTcr(x; y) = �Tcr(x; y) + To, where To is the uniform reference

or ambient temperature distribution. This type of analysis is useful for determination of a

need for thermal post-buckling analysis for a particular configuration and loading condition.

Although buckling analyses are inherently linear, and the system in concern is decidedly

nonlinear (temperature-dependent properties), there is at least one way to seek a solution within

the framework of the present formulation. Furthermore, a thermal buckling solution is very

desirable because of the insight it affords, the computational efficiency for cases when thermal

post-buckling analyses are not warranted, and the ability to provide an initial trial solution for

thermal post-buckling analyses when they are necessary. All of these points will be made more

evident in the following discussions.

Equations governing thermal buckling can be obtained from the static equations of equi-

librium, equations (4.109), and the incremental static system of equations, equation (5.11). As

stated previously, thermal buckling is a stability problem, indicating a condition for the on-

set of buckling. Therefore, the nonlinear stiffness terms due to out-of-plane deflection can be

neglected. The stiffness terms involving bending-membrane coupling can also be neglected be-

cause any increase in temperature will result in out-of-plane deflection. An exception to this

exists for antisymmetric angle-ply laminates which is a special case that will not be treated

here. Moment resultant terms are also neglected for similar reasons, i.e., the thermal load is

assumed constant through the thickness of the structure. Cases for which there is either bending-

membrane coupling or thermal moment effects should be treated with a thermal post-buckling

procedure, described in the next section, to determine the buckling temperature and static thermal

deflections. The static governing equations from (4.109), with the aforementioned restrictions,

100

simplify to the following.

��Kb 0

0 Km

��

�K�T 0

0 0

�+

1

2

�N1b 0

0 0

��Ab

Am

�=

�0

Pm�T

�(5.13)

The term[N1b] is retained because it is dependent upon the membrane displacementsfAmg only.

5.2.1 Constant Matrix Material Properties

The proposed solution to the thermal buckling problem, discussed subsequently, entails

separation of the SMA and matrix material thermal effects. Recall the SMA hybrid composite

lamina constitutive relations in principal material coordinates, after separating the SMA and

matrix thermal effects, given in equation (2.29). It is noted that the material system constituents

are not fully separated in that equation because of their combined effects in the reduced stiffness

matrix. However, the coefficientsc1a andc1m essentially render them separate in the 1–direction

and will lead to a useful form for the governing equations. Incorporation of those constitutive

relations into the finite element formulation results in a system of static governing equations

analogous to those in equations (4.109). The aforementioned assumptions inherent to thermal

buckling analysis reduce this static system to the following.

��Kb 00 Km

��

�K�T 00 0

�a

�

�K�T 00 0

�m

+1

2

�N1b 00 0

��Ab

Am

�=

�0

Pm�T

�a

+

�0

Pm�T

�m

(5.14)

The subscriptsa andm indicate SMA and matrix material, respectively. It is obvious that the

bending and membrane equations of the static problem are now de-coupled. The membrane

displacements due to an applied thermal load that varies only in the plane of the structure can

thus be determined from the equation

[Km]fAmg = fPm�T ga + fPm�T gm (5.15)

For a given applied temperature distribution�T (x; y), equation (4.56) can be used to form

the thermal force vectorsfPm�T ga andfPm�T gm and equation (5.15) can be solved for the

101

membrane displacements:

fAmg = [Km]�1(fPm�T ga + fPm�T gm)

= fAmga + fAmgm

(5.16)

The stability of this thermoelastic equilibrium configuration can be investigated through the use

of the incremental static equations (5.11).

The incremental static system of equations can be simplified as above by neglecting nonlinear

stiffness terms due to out-of-plane displacements, bending-membrane coupling stiffnesses, and

moment resultants:��

Kb 00 Km

��

�K�T 00 0

�a

�

�K�T 00 0

�m

+

�N1b 00 0

�a

+

�N1b 00 0

�m

��Ab

�Am

�=

�00

�

(5.17)

Note that the first-order nonlinear stiffness matrix due to membrane displacements has been

separated into components evaluated with membrane displacements caused by SMA and matrix

material thermal effects, respectively. Thus, the equations governing the stability of the out-of-

plane displacements can be written as

([Kb]� [K�T ]a � [K�T ]m + [N1b]a + [N1b]m)f�Abg = f0g (5.18)

Consider a temperature distribution change of the following form

�T (x; y) = ��T1(x; y) (5.19)

where� is a scalar. For a matrix material with constant material properties, the thermal force

vector and geometric stiffness matrix due to the thermal effects are linear functions of the

temperature change:

fPm�T gm�T (x;y) = �fPm�T gm�T1(x;y)

[K�T ]m�T (x;y) = �[K�T ]m�T1(x;y)

(5.20)

A similar relation can be written for the corresponding first-order nonlinear stiffness matrix

[N1b]m�T (x;y) = �[N1b]m�T1(x;y) (5.21)

102

because equations (5.15) are linear and[N1b]m is linearly dependent upon the membrane

displacements due to matrix material thermal effects.

Analogous expressions cannot be written for the thermal force vectorfPm�T ga�T (x;y),

geometric stiffness matrix[K�T ]a�T (x;y), and first-order nonlinear stiffness matrix[N1b]a�T (x;y)

associated with the SMA thermal effects over the same temperature change�T (x; y) =

��T1(x; y) as they are all nonlinear functions of temperature. The stiffness matrix[Kb] is

also nonlinear with temperature because of the influence of the SMA constituent. At the present,

let all of the terms involving the SMA thermal effects be simply evaluated at the imposed

temperature distribution and taken as constant; the rationale for this approach will be explained

subsequently. In this case, equation (5.18) can be rewritten for the temperature distribution

change�T (x; y) = ��T1(x; y) as

([Kb]� [K�T ]a � �[K�T ]m + [N1b]a + �[N1b]m)f�Abg = f0g (5.22)

which can be expressed as an eigenvalue problem

([Kb]� [K�T ]a + [N1b]a)f�g = �([K�T ]m � [N1b]m)f�g (5.23)

A critical temperature change exists if there is a solution to equation (5.23) for whichf�g is

nontrivial. The first eigen-solution gives the critical temperature change from the expression

�Tcr(x; y) = �1�T1(x; y) and the corresponding buckling mode shapef�g1.

An iterative procedure can be employed to account for the nonlinear functionality of the

terms involving the SMA thermal effects in equation (5.23). The iterative procedure entails

imposing any desired temperature distributionT1(x; y) = T0 + �T1(x; y) to evaluate the

material properties. Then, equations (2.37) and (4.56) an be used to evaluate the SMA and

matrix material thermal force vectorsfPm�T ga andfPm�T gm and the corresponding geometric

stiffness matrices[K�T ]a and [K�T ]m. Equations (5.15) can be solved for the membrane

displacementsfAmgs andfAmgm, which can then be used in equations (4.18) and (4.56) to

103

form the first-order nonlinear stiffness matrices[N1b]a and [N1b]m. Subsequently, the bending

eigenvalue problem, equation (5.23), can be formed and solved for an estimate of the buckling

temperatureT2(x; y) = �11T1(x; y) = �1

1[T0 +�T1(x; y)] and mode shapef�g1

1, where the

superscript indicates iteration number. This process is repeated until convergence is achieved

(�n1= 1:000 � :001), at which point usage of the linearized eigenvalue problem has been

rendered appropriate and accurate because�Tn(x; y) is such a small quantity that the material

properties are essentially constant.

5.2.2 Non-Constant Matrix Material Properties

The assumption that the matrix material properties were constant with temperature was

implicit to the equations and solution methodology developed in the previous section. Under

such conditions, a rationale was developed to separate the linear and nonlinear components of the

governing equations and use an iterative approach to compensate for the nonlinearity. Recall that

during the linear/nonlinear separation process, a comment was made that the reduced stiffness

matrix disallowed complete separation of the constituent material effects. This point was also

made in Chapter 2. Thus, some nonlinear effects (e.g., modulus of the SMA) do appear on

the right-hand side of equation (5.23) and the solution method of the previous section must be

applicable for at least some forms of temperature dependence in the matrix material.

This has, in fact, been observed for all of the cases considered in this study, which include

cases for which all of the matrix material properties vary significantly with temperature. The

validity of the thermal buckling solution for such cases was established by comparing with

a thermal post-buckling method. The thermal post-buckling method serves as another means

of determining a buckling temperature, as the temperature at which the thermal post-buckling

deflection departs from zero. In the event that a thermal buckling solution could not be achieved,

an initial guess would be formulated from the thermal buckling problem and used to start a

104

thermal post-buckling solution. More detail on this approach will be given following the post-

buckling solution method description at the end of the next section. A flow chart exemplifying

the thermal buckling solution procedure is shown in Figure 5.1.

Impose ∆T(x,y) = ∆Tp(x,y)

Do i=1, maxiter

Evaluate material properties &stress resultants {N∆T}s, {N∆T}m

Form [Km], {Pm∆T}s, & {Pm∆T}m

Solve for membrane displacements{Am} = [Km]-1 ({Pm∆T}s + {Pm∆T}m)

= {Am}s + {Am}m

Form [Kb], [K∆T]s, [K∆T]m,{N1b}s, & {N1b}m

Solve bending Eigenvalue problem

([Kb] - [K∆T]s + [N1b]s) {φ} = λ([K∆T]m - [N1b]m) {φ}

∆Tcr (x,y) = λ∆T(x,y)

2 1

Figure 5.1 Thermal buckling calculation flowchart. (Continued) . . .

105

2 1

∆T(x,y) = ∆Tcr(x,y)

j > maxiter?

Increase maxiter orchoose another ∆Tp(x,y)

E < crit.? ∆Tp(x,y) > ∆Tcr(x,y)?

Dynamic analysis

Store membrane deflections {Am}crand bending Eigenvector {φ}

Post-buckling analysis

If ∆T(x,y) = ∆Tp(x,y),store membrane deflections

Calculate absolutenorm error, E

No

No

No

Yes Yes

Yes

Figure 5.1 Thermal buckling calculation flowchart.

It will be shown in Chapter 8 that two buckling temperatures can exist for some SMA hybrid

composite structures. Details will be deferred to that discussion, but some general comments

are useful here. This effect is a manifestation of a material system with a SMA constituent that

has an austenite start temperatureAs greater than ambient temperature and is not to be confused

with higher-order buckling modes occurring at temperatures above initial thermal buckling. A

106

typical sequence of events in such a case can be described as follows. At temperatures less than

As, the SMA is not activated and behaves like a conventional linear thermoelastic material. So

the SMAHC structure will buckle if a critical temperature exists belowAs. The SMA becomes

activated atAs and the recovery stress rapidly increases with temperature, which overcomes the

compressive thermal stress and reduces the thermal deflection (potentially eliminating it). At

higher temperatures, the rate of increase of the recovery stress with temperature diminishes and

may be surpassed by the rate of increase in matrix compressive thermal stress. Further increases

in temperature will cause net-compressive stress development, causing the thermal deflection to

grow again or lead to a second incidence of thermal buckling for a structure that was rendered flat.

5.3 Thermal Post-buckling Analysis

Once the buckling temperature is determined, the next step will depend on the temperature

for the particular application of concern. For temperatures greater than the critical temperature

(T (x; y) > Tcr(x; y)), a post-buckling analysis must be performed to determine the thermal

deflection. The thermal post-buckling solution is started at the buckling temperatureTcr(x; y)

for expediency and ease of solution. Starting at the ambient temperature can be much more

computationally intensive (for cases with large�Tcr(x; y)) and is numerically ill-conditioned.

Further remarks about initiating the post-buckling solution will be given later in this section. For

the cases described above, where two buckling temperatures can exist, it is advisable (although

not necessary) to initiate a thermal post-buckling analysis at the first buckling temperature to

assess the evolution of the static deflection in arriving at the desired operating temperature. The

governing equations for thermal post-buckling deflection are given in equations (4.109)

�[K]� [K�T ] +

1

2[N1] +

1

3[N2]

�fAg = fP�T g (5.24)

107

where thestatic subscripts has been omitted. Note that there is no need to separate the SMA

and matrix material thermal effects in thermal post-buckling analysis. One effective approach

for solving this nonlinear system of equations is the Newton-Raphson iterative method.

Consider a static system described by equation (5.24) at the buckling temperature distribution

Tcr(x; y) and denote this equilibrium configuration byf(A)g. A small increment in the

temperature distributionT (x; y) = Tcr(x; y)+�T (x; y) will result in the formation of an adjacent

equilibrium configuration denoted byf(A+ �A)g. This adjacent equilibrium configuration can

be derived from a truncated Taylor series expansion as shown in equation (5.10):

f(A+ �A)g =�[K ]� [K�T ] +

1

2[N1] +

1

3[N2]

�fAg � fP�T g

+�[K]� [K�T ] + [N1] + [N2]

�f�Ag = f0g

(5.25)

The first collection of terms on the right-hand side of this equation is no longer a statement

of static equilibrium

f(A)g 6=�[K]� [K�T ] +

1

2[N1] +

1

3[N2]

�fAg � fP�T g 6= f0g (5.26)

as the stiffnesses and thermal load vector are evaluated atT (x; y) = Tcr(x; y) + �T (x; y), while

fAg is associated with the system at temperatureTcr(x; y). So, the expression for the adjacent

equilibrium configuration can be rewritten in terms of the tangent stiffness matrix, equation

(5.12), and an incremental load vector

[Ktan]f�Ag = f�P�T g (5.27)

where the incremental load vector is defined by

f�P�T g = fP�T g ��[K]� [K�T ] +

1

2[N1] +

1

3[N2]

�fAg (5.28)

Although a small increment in temperature may render the static system essentially linear with

temperature, the new equilibrium configuration is also governed by the geometric nonlinearity

of the first- and second-order nonlinear stiffness matrices, which are linearly and quadratically

108

dependent upon the out-of-plane displacements. Thus, an iterative procedure is necessary to

converge on the new equilibrium configuration.

The incremental form of equation (5.24) for the j-th iteration can be written as

[Ktan]jf�Agj = f�P�T gj (5.29)

The tangent stiffness and incremental load vector for the subsequent iteration

[Ktan]j+1 andf�P�T gj+1 are updated by using the updated thermal deflections

fAg = fAg+ f�Agj (5.30)

The iteration procedure is continued, with the previous iteration’s deflections as an estimate,

until the deflection estimate approaches the true equilibrium configuration. As the deflection

estimate approaches the equilibrium configuration, the expression for the incremental load vector

approaches an expression of equilibrium, at which point the incremental load vectorf�P�T gj

and, consequently, the incremental displacementsf�Agj approach zero. Convergence is achieved

when either incremental values are reduced to a specified small quantity defined by the error

criterion. A maximum normerror criterion was used in this study, as described by Bergan and

Clough [48].

As mentioned previously, the thermal post-buckling analysis is sensitive to the starting

temperature and the corresponding trial solution. An initial trial solution for the thermal post-

buckling procedure, starting at the buckling temperature, can be constructed from the thermal

buckling solution. The membrane deflections, determined from equations (5.16) evaluated at

the (first) buckling temperature, are used directly. The buckling mode shape, determined from

equation (5.23), is normalized to its maximum displacement entry and to a fraction of the

structure thickness (half-thickness was found to perform well) to give the out-of-plane portion

of the initial trial solution a physically scaled value:

fAg =

�Ab

Am

�=

�0:5hf�g=�wmax

fAmgcr

�(5.31)

109

Convergence on the displacements due to a rather large temperature increment�T (x; y)

can typically be improved by splitting the large temperature increment into a number of smaller

increments:

�Ti(x; y) =�T (x; y)

N�T

(5.32)

Iteration is performed at each of the smaller temperature increments in succession until the

converged equilibrium configuration (post-buckled static solution) is achieved for the desired

temperature. The converged displacement vector at each of the smaller temperature increments

is scaled and used as a trial vector for the next temperature increment :

fAgi= fAg

i�1

Ti(x; y)� To(x; y)

Ti�1(x; y)� To(x; y)(5.33)

Note that a mechanism for re-initialization of the out-of-plane displacements is necessary because

of the potential for elimination of the post-buckling deflection due to SMA recovery forces for

the cases with two buckling temperatures. A flow chart demonstrating the thermal post-buckling

solution procedure is shown in Figure 5.2.

One final remark concerns any potential cases for which convergence on a thermal buckling

solution is not possible. In such cases, a thermal post-buckling solution should be initiated

from the reference temperature and should proceed at temperature increments until a thermal

post-buckling deflection is developed. At each increment in temperature, the in-plane problem

of the buckling solution should be solved for the in-plane deflections. The in-plane deflections

will be used to form the thermal buckling eigenvalue problem. The eigenvector resulting from a

single eigen-solution would then be used, along with the in-plane deflections, to form an initial

guess for the post-buckling solution at that temperature. Once a thermal post-buckling deflection

is achieved, a refined estimate of the buckling temperature can be determined by restarting the

post-buckling solution in the vicinity of the first solution with a smaller temperature increment.

110

2 1

Divide total ∆T(x,y) intoN∆T smaller increments

∆Ti(x,y) = (Tp(x,y) - Tcr(x,y))/N∆T

Form initial trial vectorfrom buckling solution

{A}0 ={ 0.5h {φ}/φwmax}{Am}cr

T0(x,y) = Tcr(x,y)

Do i = 1, N∆T

Ti(x,y) = Ti-1(x,y) + ∆Ti(x,y)

{A}i = {A}i-1 Ti(x,y)-To(x,y)

Ti-1(x,y)-To(x,y)

Scale trial vector for Ti(x,y)

Evaluate material properties &stress resultants {N∆T}

Figure 5.2 Thermal post-buckling calculation flowchart. (Continued) . . .

111

2 1

Do j=1, maxiter

Form tangent stiffness

[Ktan]j = ([K] - [K∆T] + [N1] + [N2])and incremental load vector

{δP∆T}j = {P∆T} - ([K] - [K∆T] + 1/2[N1] + 1/3[N2]) {A}i

j

Solve for incremental displacements{δA}j = [Ktan]-1 {δP∆T}j

Increment trial vector{A}i = {A}i + {δA}j

Calculate max norm error, E

i > N∆T?

{Ab}i = {Ab}0

{Ab}i ~ 0?

j > maxiter?E < crit.?

Dynamic analysis

No

No

No

No

Yes YesYes

Yes

Increase maxiter or N∆T

~

Figure 5.2 Thermal post-buckling calculation flowchart.

112

5.4 Analytical Validation

In the previous sections, a formulation and approach for prediction of the static stability

(thermal buckling) and large-deflection (thermal post-buckling) of SMA hybrid composite panel-

type structures was derived. The formulation and implementation will now be validated through

comparison with fundamental cases and experimental measurements. Comparisons will be made

with known solutions for the cases of a simply-supported isotropic beam, simply-supported and

clamped isotropic plates, and a simply-supported symmetrically-laminated composite plate to

benchmark the analysis. Comparisons will be made with experimental measurements for the

case of a clamped-clamped SMAHC beam to fully validate the thermomechanical model for

static response.

Note that the effects of initial stresses, initial deflections, thermal moments due to a

temperature gradient through the thickness, and bending-membrane coupling (non-symmetric

composite laminates) are not included in the analysis, although solution with such conditions

is quite tractable by the thermal post-buckling solution technique. Therefore, the thermal

buckling cases presented are true bifurcation problems and, as such, the resulting post-buckling

deflection could be positive or negative. In practice, the predicted buckling direction is typically

predetermined by the sign of the initial guess. The sense of the buckling deflection in a particular

experimental configuration is usually influenced by some combination of imperfections including

initial stresses, initial deflections, and anomalies in boundary conditions. The positive sense will

be taken in the purely numerical results to be presented and the sense will be chosen to match

the experimental behavior when comparing with experimental measurement.

5.4.1 Simply-Supported Isotropic Beam

Consider a simply-supported aluminum beam with dimensions of 0.5588x0.0254x0.0023 m

(18x1x0.090 inches) and subject to a uniform temperature increase. The material properties for

113

Table 5.1 Material properties for aluminum specimens.

Young’s Modulus 68.95 GPa 10 Msi

Shear Modulus 26.55 GPa 3.85 Msi

Poisson’s Ratio 0.3

Coeff. of Thermal Exp. 22.3��/�F 12.4��/�F

the specimen are shown in Table 5.1.

A classical solution for the critical temperature change to induce buckling can be easily

derived based upon beam-column theory, as shown by Boley and Weiner [49] for example. For

a uniform temperature distribution, the axial load is given by

PT = AE��T (5.34)

whereA is the beam cross-sectional area,E is the Young’s modulus, and� is the thermal

expansion coefficient. Thermal buckling occurs when the axial load reaches the Euler buckling

load, defined by the familiar expression

Pcr =�2EI

L2(5.35)

whereA is the area moment of inertia, andL is the beam length. Thus, the critical temperature

change is found to be

�Tcr =�2I

�AL2(5.36)

This gives a numerical value of 1.658�F for the predicted critical temperature change from

beam-column analysis as compared to 1.643�F for the present finite element analysis with a

36x2-element mesh. The slight discrepancy is attributable to the fact that the classical analysis

does not include transverse effects because it is a one-dimensional analysis.

114

5.4.2 Simply-Supported and Clamped Isotropic Plates

A classical solution for the critical temperature change to induce buckling for an simply-

supported isotropic plate is also given by Boley and Weiner [49]. For a uniform temperature

distribution, the in-plane thermal force is given by

NT = �E

h

2Z

�

h

2

�T dz = �Eh�T (5.37)

whereh is the thickness of the plate. Thermal buckling occurs when the in-plane thermal load

reaches the reaches a critical value, given by

NT cr = (1� �)

�1 +

a2

b2

��2D

a2(5.38)

where� is the Poisson’s ratio,a and b are the panel length and width, andD is the flexural

rigidity given by

D =Eh

3

12(1 � �2)(5.39)

Thus, the critical temperature change is given by the following expression

�Tcr = (1� �)

�1 +

a2

b2

��2D

a2�Eh(5.40)

As an example, consider a simply-supported aluminum plate with dimensions of

0.3556x0.254x0.001 m (14x10x0.040 inches) and the material properties given in Table

5.1. The classical solution for this case gives a numerical value of 1.233�F for the predicted

critical temperature change, which is matched exactly by the present finite element analysis

with a 14 x 10 element mesh.

A classical solution for the thermal post-buckling deflection of a clamped isotropic plate

was developed by Paul [50]. He used a twenty-five term series approximation in a classical

formulation with von Karman deformation assumptions. One case he analyzed consisted of

115

Dimensionless Thermal Load

Nor

mal

ized

Def

lect

ion

,Wm

ax/h

0 1 2 3 4 5 6 7 8 9 10 11 120

0.25

0.5

0.75

1

1.25

1.5

1.75

2

ClassicalFinite Element

Figure 5.3 Normalized maximum deflection versus dimensionless thermal load for a clamped,0.102x0.102x0.001 m (4x4x0.040 inch), aluminum plate with uniform thermal load.

a clamped aluminum plate with dimensions of 0.102x0.102x0.001 m (4x4x0.040 inches) with

material properties as given in Table 5.1, except with a slightly different thermal expansion

coefficient�=22.5 ��/�C (12.5��/�F) and a uniform thermal load. A plot of the normalized

center deflection versus dimensionless thermal load for this case is shown in Figure 5.3. It

can be seen that the present finite element analysis is in excellent agreement with the classical

solution. The fine resolutionN�T = 100 for the finite element results was used for presentation

116

purposes only. Convergence on the solution at the final temperature was readily achieved

with only one temperature step. In fact, the solution was much more expedient albeit not

conducive to conveying information about the deflection behavior at intermediate temperatures.

The dimensionless thermal load shown as the abscissa of the plot is given by Paul as

Load= 12(1 + �)��Ta2

�2h2(5.41)

The range on the load from approximately 5.3 to 12 is equivalent to a�T from zero to 15.6�C

(60�F).

5.4.3 Simply-Supported Composite Plate

Meyers and Hyer [51] employed a Rayleigh-Ritz method to developed a classical solution

for the buckling temperature and the thermal post-buckling deflection of simply-supported,

symmetrically-laminated composite plates. They analyzed plates with fixed and sliding simple

supports. For example, Meyers and Hyer reported a buckling temperature of 20.8�C (69.4�F)

for a graphite-epoxy plate with dimensions of 0.152x0.152x0.001 m (6x6x0.040 inches), quasi-

isotropic lamination of (±45/0/90)s, fixed simple supports, a uniform temperature distribution,

and material properties as shown in Table 5.2. The corresponding solution from the present

finite element analysis is 20.6�C (69.05�F) with a 6 x 6-element mesh.

Table 5.2 Material properties in principal material coordinates for a graphite-epoxy lamina.

Young’s Modulus, E1 155.1 GPa 22.5 Msi

Young’s Modulus, E2 8.07 GPa 1.17 Msi

Shear Modulus, G12 4.55 GPa 0.66 Msi

Poisson’s Ratio,�12 0.22

Coeff. of Thermal Exp.,�1 -0.07��/�C -0.04��/�F

Coeff. of Thermal Exp.,�2 30.1��/�C 16.7��/�F

117

∆T/∆Tcr

Nor

mal

ized

Def

lect

ion

,Wm

ax/h

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Figure 5.4 Normalized maximum deflection versus normalized temperaturechange for a simply-supported, 0.152x0.152x0.001 m (6x6x0.040 inch),

(±45/0/90)s graphite-epoxy laminate with uniform thermal load.

Corresponding thermal post-buckling deflections predicted by the classical and finite element

analyses are shown in Figure 5.4. In this case, normalized center deflection is plotted against

normalized temperature change. Excellent agreement is achieved between the classical and finite

element analyses. Again, a fine resolutionN�T = 100 was used in the finite element analysis

for presentation purposes, but more expedient calculation of the final thermal post-buckling

deflection was achieved with only one temperature step.

118

Recall that both cases for which thermal post-buckling deflections were calculated are

classified as bifurcation problems. The analogous negative deflection results could be attained by

using a thermal buckling eigenvector of opposite sign in the formation of an initial trial vector.

Note that this ambiguity is of no consequence to the thermal buckling solution as it is the relative

magnitudes in the buckling eigenvector that are important.

5.5 Experimental Validation

The thermal buckling and post-buckling solutions were validated against known solutions

for conventional structures in the previous section. Attention will now turn to experimental

measurement of the thermoelastic response of a SMAHC beam. The experimental results will

be used to demonstrate some of the phenomena discussed earlier in this chapter and to validate

the formulation and solution methods for the case of a SMAHC structure.

5.5.1 Clamped SMAHC Beam

One of the SMAHC beam specimens is shown mounted in an aluminum fixture designed to

provide clamped boundary conditions for beam specimens with this geometry in Figure 5.5. The

beam has an unsupported length of 0.4572 m (18 in) within the mechanical grips. The fixture was

also designed to provide electrical connection and mechanical restraint for the Nitinol ribbon leads

protruding from the ends of the SMAHC beam specimens. The mechanical and electrical grips

are indicated in the figure by the symbols “m” and “e”, respectively. The electrical connections

are isolated from the fixture by a layer of fibrous ceramic insulation. The beam is also thermally

isolated from the fixture by a layer of this insulation on both sides. The beam is heated by DC

electrical current controlled by a thermocouple measurement located approximately on the beam

centerline and 0.0127 m (0.5 in) from the right mechanical grip, indicated in the figure by “tc”.

The thermal controller consists of a process controller (Omega model CN77353–PV), and a DC

119

m e

SMAHC Beam

Fixture

Power Cables

tc

tc

tc

Figure 5.5 Front view of the SMAHC beam specimen mountedin clamping fixture used for static and dynamic testing.

power supply (HP model 6652A). Thermocouples also monitor the temperature at each end of

the fixture, indicated in Figure 5.5, and one measures the ambient air temperature.

A rear view of the experimental configuration is shown in Figure 5.6, which shows the

installation of a laser displacement transducer. The laser displacement transducer (Aromat

model LM100) was attached to a three-axis optical positioning traverse, and the assembly was

mounted in the vicinity of the beam mid-span. The beam mechanical grips and the ribbon

mechanical/electrical grips were torqued to 13.56 N•m (120 in•lbf). The three-axis traverse

was used to position the laser impingement position at the beam mid-span and to adjust the

laser transducer/beam surface distance to produce zero offset. Changes in beam and fixture

temperatures during the gripping process and prior to starting the thermal controller were

insignificant. The beam was heated from ambient temperature to 121.1�C (250�F) over a duration

of 5 minutes and 11 seconds, then allowed to cool naturally to ambient temperature again. The

overall duration of the test was 13 minutes and 26 seconds. Measurements of the ambient air,

right/left fixture end, and beam temperatures and the laser transducer output were recorded at

approximately 2.5 second intervals.

The beam was modeled with a 36x2 mesh of the 24 degree-of-freedom rectangular plate

120

SMAHC Beam

3-Axis Positioner

Fixture

Laser DisplacementTransducer

Figure 5.6 Experimental configuration for measurement ofthe static thermoelastic response of a SMAHC beam specimen.

elements described in Chapter 4. This test constituted thermal cycle number four (4) on the

Nitinol ribbon in this particular beam specimen. Element properties were constructed from

the appropriate measured material properties described in Chapter 3 and given in Tables 3.5

and 3.8. Although the beam ends are held firmly within the mechanical grips, the boundary

conditions are not well approximated by perfectly clamped conditions. It will be shown in the

Experimental Validation section of Chapter 6 that it was necessary to alter the clamped boundary

conditions with torsional springs, with a spring rate of 1.98 N•m/rad (17.5 in•lbf/rad), applied to

the longitudinal-slope degrees of freedom to match the fundamental natural frequency. It will

be assumed that the boundary conditions for the thermal post-buckling solution are essentially

the same as that for the dynamic analysis. In previous experiments, the temperature distribution

generated under these conditions was measured with an infrared thermal imaging camera. It was

found that the temperature distribution was remarkably uniform along the unsupported length of

the beam. Therefore, the thermal load was modeled as a uniform temperature distribution.

A comparison between the measured and predicted thermal post-buckling deflection (nor-

malized to the beam thickness) at the beam mid-span is shown in Figure 5.7. Note that the

121

measured deflection does exhibit the first part of the complex thermoelasitic behavior described

earlier in this chapter; buckling at a temperature belowAs, achieving a maximum deflection

where the Nitinol recovery stress starts to dominate, and returns to a flat configuration for the

remainder of the test. The reason that a second instance of buckling is not exhibited is that the

volume fraction of Nitinol is high enough in this case to drive the predicted second buckling

temperature well beyond the glass-transition temperature for the matrix material.

Temperature, oC

Nor

mal

ized

Def

lect

ion,

W/h

20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

MeasurementPrediction

Figure 5.7 Measured and predicted thermal post-buckling deflection(normalized to the beam thickness) at the mid-span of the SMAHC beam.

122

The predicted post-buckling behavior is in excellent qualitative agreement with the experi-

mental measurement. The main differences between the predicted and measured responses are

the slope of the curve from initial buckling to the maximum deflection point, the magnitude of

the maximum deflection, and the temperature at which the post-buckling deflection is returned to

zero. These discrepancies are attributable to the fact that these systems are extremely sensitive

to a multitude of factors including initial deflection, initial stress, boundary conditions, etc. For

example, initial deflection in the physical beam is the reason for the more gradual growth of

thermal deflection in the experimental results. The discrepancies in peak deflection and deflection

elimination temperature are attributable to boundary conditions.

Note that the in-plane boundary conditions are modeled as immovable. Recall that in the

physical system the beam continues through the mechanical grips and the Nitinol leads are

terminated beyond that in the electrical grips. Only the ribbon/matrix interface supports the

recovery force within the mechanical grip area. The main restraint for the Nitinol is the

mechanical termination at the electrical grips. Thus, the compliance of the entire support

structure, from one electrical grip to the other, needs to be considered. It was found that

the layer of electrical insulation between the electrical and mechanical grips was the cause for

the discrepancies. A compression test was performed on the insulating material and the modulus

was found to be less than 137.9 MPa (20 ksi). While the exact stiffness of the boundary would

be difficult to quantify, there is no doubt that strain in the insulation is partially relieving the

recovery stress and causing a diminished stiffening effect in the physical system. Changes

in the model to reflect the compliant in-plane boundaries would move the predicted buckling

temperature, peak deflection, and deflection elimination temperature to better agreement with

experimental measurement.

In this chapter, the static stability of a structure described by the governing equations

developed in the previous chapter was analyzed and solution procedures were developed to

123

predict the thermal buckling and post-buckling behavior of SMAHC structures. The formulation

and procedures were validated against known solutions prior to attempting correlation with

experiment. Experimental results from a SMAHC beam specimen confirm complex thermoelastic

behavior that was predicted by the model. Excellent agreement was achieved between prediction

and experimental measurement and reasons for existing discrepancies were discussed. A similar

approach will be taken in the next chapter, where the dynamic response performance of a

SMAHC beam will be demonstrated and compared with numerical prediction.

124

Chapter 6Dynamic Response

The analytical techniques and computational methods used to solve the equations governing

the dynamic response of SMA hybrid composite panel-type structures subjected to mechanical

and thermal loads are treated in this chapter. The thermal load is assumed to be steady-state

and uniform through the thickness of the structure, i.e.,T = T (x; y). Mechanical loads are

assumed to consist of surface pressure, concentrated forces, and inertial (base acceleration)

loads. Continuous and discrete frequency domain analyses are presented. Validation of the

analyses with known solutions for fundamental cases and with experimental results for several

cases including a SMA hybrid composite beam are presented.

6.1 Equations of Motion

The equations of motion for the system described above were derived in Chapter 4, equation

(4.109), and can be rewritten in the compact form

[M ]f �Ag+��K�fAg = fP (t)g+ fF (t)g � [M ]f �Ugg (6.1)

where��K�

is the stiffness matrix for the flat or buckled/deflected structure. For a flat (unbuckled)

structure ��K�= [K]� [K�T ] + [N1] unbuckled

��K�= [Ktan] = [K]� [K�T ] + [N1] + [N2] buckled

(6.2)

Note that the only nonzero entries in[N1] for the unbuckled case are due to[N1b] resulting

from the in-plane displacements, see equation (5.16), and the stiffness matrix for the thermally

125

buckled structure is precisely the converged tangent stiffness matrix from the thermal post-

buckling solution. For a thermally buckled/deflected structure

��K�= ([K]� [KN�T ] + [N1] + [N2]) (6.3)

where the nonlinear stiffness matrices[N1] and [N2] are evaluated using the thermally buck-

led/deflected displacementsfAgs. Although it is not necessary, the three dynamic load vectors

will be considered separately for clarity in the dynamic response solution and because practical

experimental configurations dictate separate loading cases.

6.1.1 Acoustic Excitation

Recall the form for the surface pressure excitation load vector from equation (4.57)

fP (t)g =

�Pb(t)

0

�(6.4)

The bending components for the system are assembled from the element contributions given in

equations (4.56), which can be rewritten as

fpb(t)g =

ZAe

p(�; �; t)fLw(�; �)g dAe (6.5)

wherep(�; �; t) is the spatial and temporal description of the pressure,fLw(�; �)g is the vector

of out-of-plane interpolation functions,(�; �) are the coordinates in the plane of the structure,

and Ae is the area of the element.

In this study, it is assumed that the surface pressure is generated by acoustic excitation. Then

the pressure loading on the structure consists of the incident pressure, the reflected pressure, and

the transmitted pressure as shown schematically in Figure 6.1:

p(�; �; t) = pi(�; �; t) + pr(�; �; t)� pt(�; �; t) (6.6)

The pressure loading may also be written as the sum of theblockedpressure and theradiated

pressure, as suggested by Roussos [52]. The blocked pressure is the pressure on the incident side

126

z η,y

pipr

θi

φi

pt

ξ,x

Figure 6.1 Schematic of acoustically excited panel.

when the structure is considered rigid, i.e.,pb(�; �; t) = 2 pi(�; �; t), and the radiated pressure

is that due to structure motion only. The acoustic radiation problem can be solved subsequent

to the forced response solution by assuming that the radiated pressure is negligible compared to

the blocked pressure. This assumption results in inaccurate predictions only at frequencies near

the panel fundamental resonance. Thus, the load vectorfPb(t)g is formed using the blocked

pressure (p(�; �; t) = pb(�; �; t) = 2pi(�; �; t)) and the transmitted pressure (pressure radiated

to the free field) can be determined from the resulting structural motion, without the need for

a fully coupled analysis.

Furthermore, it is assumed in this study that the acoustic pressure excitation consists of

obliquely incident plane waves of arbitrary temporal variation. Then, the general form for the

acoustic pressure along the direction of propagation given by a Cartesian coordinates has the

formpi(�; �; t) = p(t� s=c)

= p(t� n � ~x=c)(6.7)

where n is a unit normal aligned with the direction of propagation in the coordinate system

designated by~x. So, the incident pressure (i.e.,z = 0) can be described by

pi(�; �; t) = p(t� � sin �i cos �i=c � � sin �i sin�i=c) (6.8)

127

where�i and�i are the polar and azimuthal angles of incidence, respectively.

The incident pressure is modeled by nodal pressures because of the spatial discretization of

the finite element approach. Then, the blocked pressure element load vector in equation (6.5)

becomes

fpb(t)g =

Z

Ae

2 fLw(�; �)gp(t� � sin �i cos �i=c� � sin �i sin�i=c) dAe (6.9)

A more simple form for non-normal blocked acoustic pressure loading will be developed in the

discrete spectral response section. In the case of normal acoustic pressure excitation, equation

6.9 simplifies to the following

fpb(t)g =

Z

Ae

2 fLw(�; �)gp(t) dAe

= f bgp(t)

(6.10)

So, the acoustic load vector for the entire system is assembled from element contributions given

in either of the latter two equations. The system load vector for the normal acoustic excitation

case takes a simple form

fPb(t)g = f�gp(t) (6.11)

6.1.2 Concentrated Force Excitation

The concentrated force load vector from equation (4.57) has an analogous form as follows

fF (t)g =

�Fb(t)

0

�(6.12)

The out-of-plane displacement (bending) components for the system are assembled from the

element contributions given in equations (4.56), which can be rewritten as

ffb(t)g =

nfXi=1

ZAe

f 0

i(t) �(f�g � f�gi)fLw(�; �)g dAe (6.13)

wheref 0

i(t) is the temporal description of the concentrated force(s),�(f�g � f�gi) is a spatial

Dirac delta function, andnf is the number of forces within the element area. Note that the

concentrated forces can be arbitrarily correlated.

128

It is assumed that the concentrated force(s) are applied at nodal locations in the model.

Otherwise, the effect of the concentrated force would be distributed to nodes of the element

containing the force, which is an undesirable effect. Evaluation of the element area integral

in equation (6.13) simplifies the form for the element contributions to the concentrated force

load vector:

ffb(t)g =

nfX

i=1

f 0

i(t)fLw(�i; �i)g

=

nfX

i=1

f bgif0

i(t)

(6.14)

Assembly of the element contributions in the system concentrated load vector simply positions

each concentrated force at a location in the system load vector corresponding to the associated

out-of-plane degree of freedom. Then, the resulting concentrated load vector for the entire

system can be written as follows

fFb(t)g =�f 0(t)

(6.15)

where the only nonzero components inff 0(t)g correspond to forced nodal locations.

6.1.3 Inertial Excitation

The inertial loading can be expressed in an analogous load vector form from equation (4.57):

fG(t)g =

�Gb(t)0

�= �

�Mb 00 Mm

��Ug(t)0

�(6.16)

The out-of-plane displacement components for the system are assembled from the element

contributions:fgb(t)g = �[mb]f�ug(t)g

= �[mb]f1g�ug(t)

= f bg�ug(t)

(6.17)

wheref1g is a vector with unit entries and�ug(t) is the temporal description of the spatially

uniform base acceleration. So, the system inertial load vector can be written as follows

fGb(t)g = f�g�ug(t) (6.18)

129

Solutions to the system of equations given in equation (6.1) can be sought using physical

or modal degrees of freedom. The modal approach has the advantage of vastly improved

computational speed without significant loss in accuracy in many cases. The modal approach

is also quite useful, particularly when comparing with experimental measurements, as powerful

tools such as experimental modal analysis can be used for incremental validation of the analytical

model. The modal approach will be used in this study and is discussed briefly here.

6.2 Eigen-solution and Modal Coordinates

Natural frequencies!r and mode shapesf�rg of the vibration about static equilibrium can

be obtained from the dynamic eigenvalue problem

!2

r [M ]f�rg =��K�f�rg (6.19)

A set of uncoupled modal equations with reduced degrees of freedom can thus be obtained from

equation (6.1) as

dIcf�q(t)g+ d2�r!rcf _q(t)g+ d!2

r cfq(t)g = ff(t)g (6.20)

where r = 1; 2; . . . ; Nq, the delimitersdc indicate a diagonal matrix, anddIc is the identity

matrix, and�r is the modal critical damping ratio. The modal force vectorff(t)g is given by

one of the following expressions as the time dependent loads are considered separately

ff(t)g = [�]T fp(t)g acoustic pressure excitation

= [�]T f�gp(t) normal acoustic pressure

ff(t)g = [�]T�f 0(t)

concentrated force excitation

ff(t)g = [�]T f�g�ug(t) inertial/base excitation

(6.21)

The physical degrees of freedom can be recovered from the truncated modal transformation:

fA(t)g =

NqX

r=1

f�grqr(t) = [�]fq(t)g Nq � Ndof (6.22)

130

where[�] is the modal matrix,Nq is the number of modes retained, andNdof is the total number

of degrees of freedom.

The modal response, solution to the modal equations of motion (6.20 ), can be determined by

time domain or frequency domain analysis. In either case, a continuous or discrete approach can

be employed. In this study, dynamic analysis is done in the frequency domain through continuous

and discrete techniques. The continuous frequency domain approach is included mainly for

demonstrative and comparative purposes. The discrete frequency domain approach employed

in this study is the main spectral solution tool because of the ability to handle complicated

frequency distributions in a straightforward manner.

6.3 Continuous Spectral Analysis — Random Response

The time response of the modal coordinates can be obtained from the convolution

(Duhamel’s) integral

fq(t)g =

tZ

t0

dh(t� � )cff(� )g d� =

t�t0Z

0

dh(� )cff(t� � )g d� (6.23)

wheredh(t)c is a diagonal matrix of impulse response functions andff(t)g is the modal force

vector. The modal response correlation matrix[Cqq(t1; t2)] can be formed from the following

expected value

E[fq(t1)gfq(t2)gT ] =

t1�t0Z

0

t2�t0Z

0

dh(�1)cE[ff(t1 � �1)gff(t2 � �2)gT ]dh(�2)c d�1d�2 (6.24)

where the expected value in the integrand is the correlation matrix of the excitation�Cff(t1 � �1; t2 � �2)

�.

The correlation matrix forstationary excitationbecomes a function of the time difference

only and will be denoted as follows

E[ff(t1 � �1)gff(t2 � �2)gT] =

�Cff(t2 � t1 + �1 � �2)

�(6.25)

131

For a load with spectral densitySff(!) or Gff (f), the Wiener-Khinchine relations for stationary

random processes gives

�Cff (t2 � t1 + �1 � �2)

�=

1Z�1

�Sff(!)

�ei!(t2�t1+�1��2) d!

=

1Z

0

�Gff (f)

�ei2�f(t2�t1+�1��2) df

(6.26)

whereGff (f) = 4�Sff(!) for ! = 2�f (f � 0). The modal response correlation matrix,

equation (6.24), then becomes

[Cqq(t1; t2)] =

1Z�1

t1�t0Z

0

dh(�1)cei!�1d�1

�Sff(!)

�ei!(t2�t1)

t2�t0Z

0

dh(�2)ce�i!�2d�2 d!

=

1Z

�1

dH�(!; t1)c�Sff(!)

�dH(!; t2)ce

i!(t2�t1)d!

(6.27)

wheredH(!; t)c is a diagonal matrix of transient frequency response functions.

For stationary response, let t0 ! �1 andH(!; t)! H (!) so the modal displacement

correlation matrix becomes

[Cqq(� )] =

1Z�1

dH�(!)c�Sff(!)

�dH(!)cei!�d! (6.28)

where � = t2 � t1 and

Hr(!) =1

!2r � !2 + 2i�r!r!

(6.29)

is a stationary frequency response function. Assuming mean-square differentiability, Correlations

of the modal velocity and acceleration responses can be determined by assuming mean-square

differentiability:

[C _q _q(� )] =

1Z�1

!2dH�(!)c�Sff(!)

�dH(!)cei!� d! (6.30)

and

[C�q�q(� )] =

1Z

�1

!4dH�(!)c�Sff(!)

�dH(!)cei!� d! (6.31)

132

where the relations@@t1

= �

dd�

and @@t2

= dd�

have been employed. The three modal response

correlation matrices can be written in the following concise form.

([Cqq(�)]; [C _q _q(� )]; [C�q�q(� )]) =

1Z

�1

!(0;2;4)dH�(!)c�Sff(!)

�dH(!)cei!� d! (6.32)

Note that an arbitrary element(r; s) of the modal response correlation matrices has the following

form

(E[qr(t)qs(t+ � )]; E[ _qr(t) _qs(t+ � )]; E[�qr(t)�qs(t+ � )]) = E[Cqrqs(� ); C _qr _qs(� ); C�qr �qs(� )]

=

1Z�1

!(0;2;4)H�

r (!)Hs(!)Srsff (!)e

i!�d!

=

1Z�1

(2�f)(0;2;4)H�

r (f)Hs(f)Grsff (f)e

i2�f�df

(6.33)

Recall the expression for the modal force vectorff(t)g for each load type from equations

(6.21). Then, the modal force vector correlation and spectral density matrices become

�Cff(� )

�= [�]

T[Cpp(� )][�] )

�Sff (!)

�= [�]

T[Spp(!)][�]

�Cff(� )

�= [�]T

�Cf 0f 0(� )

�[�] )

�Sff (!)

�= [�]T

�Sf 0f 0(!)

�[�]

�Cff (� )

�= [�]

Tf�gC�ug�ug(� )f�g

T[�] )

�Sff (!)

�= [�]

Tf�gS�ug �ug(!)f�g

T[�](6.34)

for acoustic pressure, concentrated force, and base acceleration excitation, respectively. For the

case of normal acoustic pressure, the corresponding relations above reduce to

�Cff (� )

�= [�]Tf�gCpp(� )f�g

T [�] )�Sff (!)

�= [�]T f�gSpp(!)f�g

T [�] (6.35)

The main simplification in these expressions is the presence of a single auto-correlation/power-

spectral-density function rather than a covariance/spectral-density matrix describing the spatial

variation of the acoustic load. Note that for the cases of normal acoustic, single concentrated

force, or base acceleration excitation, the spectral density function in the integrand of equation

133

(6.33) is the same for all elements(r; s) of the modal response correlation matrices:

(E[qr(t)qs(t+� )]; E[ _qr(t) _qs(t+�)]; E[�qr(t)�qs(t+�)]) =

1Z

�1

!(0;2;4)H�

r (!)Hs(!)S��(!)ei!�d!

=

1Z

0

(2�f)(0;2;4)

H�

r (f)Hs(f)G��(f)ei2�f�df

(6.36)

Examples of the evaluation of these integrals for determination of the dynamic response quantities

are given in Appendix F.

The responses in the physical coordinates can be recovered from the modal expansion,

equation (6.22), i.e.,

[CAA(� )] = [�][Cqq(� )][�]T (6.37)

Thus, the physical coordinate displacement, velocity, and acceleration correlation matrices

become

�[CAA(� )];

�C _A _A(� )

�;�C �A �A(� )

��=

1Z�1

!(0;2;4)[�]dH�(!)c�Sff (!)

�dH(!)c[�]

Tei!� d!

(6.38)

6.4 Discrete Spectral Analysis

A discrete spectral analysis approach will now be formulated as an alternative to the

continuous approach because continuous approaches are limited to rather simple loading cases,

which are not very realistic for practical applications and not easily generated in an experimental

situation. In practice, there is always a finite amount of data. Thus, one would employ a Finite

Fourier Transform to a sampled time history as observed through a time window. Furthermore,

for discretely sampled time data, a Discrete Fourier Transform (DFT) can be used to estimate

the Finite Fourier Transforms, etc. A particularly efficient means of performing a DFT is known

as the Fast Fourier Transform (FFT). Of greater importance is that most experimental data is

134

processed via the FFT. A means of using experimentally measured data has obvious beneficial

implications.

A discrete spectral analysis can be employed by assuming that the modal force vector, and

thus the modal response, can be written as the sum of harmonic functions:

ff(t)g =N!�1X

n=0

ffngei!nt (6.39)

and

fq(t)g =N!�1X

n=0

fqngei!nt (6.40)

whereN! is the number of discrete frequencies in the transformation andffng andfqng are the

complex spectral amplitude vectors, one for each frequency (see Doyle [53]). Recall the modal

expansion for the physical coordinate response from equation (6.22):

fA(t)g =N!�1X

n=0

fAngei!nt

=N!�1X

n=0

[�]fqngei!nt

(6.41)

Substitution of equations (6.39) and (6.40) into the modal equations of motion, equation (6.20),

givesN!�1X

n=0

(�!2ndIc+ i!nd2�r!rc+ d!2rc)fqngei!nt =

N!�1X

n=0

ffngei!nt (6.42)

This relation can be written for any one of the discrete frequencies!n as

�d�!2nc+ di2�r!r!nc + d!2rc

�fqng = ffng (6.43)

This relation is simply a set of uncoupled algebraic equations for the modal displacement spectral

amplitudes. Thus, the modal displacement spectral amplitude vector becomes

fqng = dH(!n)cffng (6.44)

where thedH(!n)c is a diagonal matrix of frequency response functions given by

Hr(!n) =1

!2r � !2n + i2�r!r!n(6.45)

135

Recall the expression for the incident acoustic pressure in equation (6.8). In this discrete

spectral analysis, the incident pressure is written as a sum of harmonic components:

pi(�; �; t) = p(t� � sin �i cos �i � � sin �i sin�i)

=N!�1X

n=0

pnei(!nt�kn� sin �i cos�i�kn� sin �i sin�i) (6.46)

wherekn = !n=c is the acoustic wave number. Then, the blocked pressure element load vector

from equation (6.9) becomes

fpb(t)g =N!�1X

n=0

Z

Ae

2pnfLw(�; �)gei(!nt��n��n)

=N!�1Xn=0

f bngpnei!nt

(6.47)

where�n = kn sin �i cos�i and�n = kn sin �i sin�i and f bng contains the integrated spatial

correlation information of the incident acoustic pressure. The assembled system load vector due

to acoustic excitation would be written as

fPb(t)g =N!�1Xn=0

f�ngpnei!t (6.48)

The modal force vector is related to the mechanical excitation load vectors from the Galerkin

weighted average of the modal system (modal system diagonalization), equation (6.21):

ff(t)g = [�]T fp(t)g ) ffng = [�]T f�ngpn

ff(t)g = [�]T�f 0(t)

) ffng = [�]T ff 0ng

ff(t)g = [�]T f�g�ug(t) ) ffng = [�]T f�gugn

(6.49)

wherepn, f 0n, andugn are the complex spectral amplitudes of the acoustic pressure, concentrated

force, and base acceleration, respectively. Note that, in contrast to the acoustic pressure modal

force spectral density matrix in the continuous spectral analysis, the expression for the modal

force spectral amplitude due to acoustic pressure is valid for any angles of incidence.

If the excitation is deterministic, the dynamic response solution in terms of modal dis-

placement, velocity, and acceleration time responses can be computed by an inverse FFT from

136

equations (6.40) and (6.44):

(fq(t)g; f _q(t)g; f�q(t)g) =N!�1X

n=0

(i!n)(0;1;2)dH(!)cffnge

i!nt (6.50)

where the modal force spectral amplitude has one of the forms given in equations (6.49). Then

the responses in physical coordinates follow from equation (6.41):

�fA(t)g; f _A(t)g; f �A(t)g

�=

N!�1Xn=0

(i!n)(0;1;2)[�]dH(!)cffnge

i!nt (6.51)

However, for stochastic excitation, a statistical approach must be used. Additionally, dynamic

response cross-correlation quantities are of interest for the acoustic radiation discussion to follow

in the next chapter. Therefore, the necessary relations for stochastic dynamic response will be

developed next.

The modal displacement response correlation matrix can be formed from equation (6.40) as

the following expected value

[Cqq(t1; t2)] = E[fq(t1)gfq(t2)gT]

=N!�1Xm=0

N!�1Xn=0

E[fq�mgfqngT ]ei(!nt2�!mt1)

(6.52)

Substituting from equation (6.44) gives

[Cqq(t1; t2)] =

N!�1Xm=0

N!�1Xn=0

dH�(!m)cE[ff�mgffngT ]dH(!n)ce

i(!nt2�!mt1) (6.53)

whereE[ff�mgffngT ] is a generalized spectral matrix. Note that if the excitation is deterministic,

the ensemble average indicated by the expected value in equation (6.53) is not necessary. If the

excitation is stochastic, the necessary ensemble averages can be performed ahead of time.

For a stationary process, the expected value of the excitation complex spectral amplitudes

takes a special form [54]

E[ff�mgffngT ] = E[ff�mgffng

T ]�mn (6.54)

137

where�mn is the Kronecker delta. Then, the stationary modal displacement correlation matrix

results

[Cqq(� )] =

N!�1X

n=0

E[fq�ngfqngT ]ei!n� (6.55)

where� = t2 � t1. The stationary modal velocity and acceleration correlation matrices follow

from the relations @@t1

= � dd�

and @@t2

= dd�

as

[C _q _q(� )] =N2�1X

n=0

!2nE[fq�ngfqng

T ]ei!n�

[C�q�q(� )] =

N2�1X

n=0

!4nE[fq�ngfqng

T]ei!n�

(6.56)

The full expressions can be written in the following concise form for convenience

([Cqq(� )]; [C _q _q(� )]; [C�q�q(� )]) =N!�1X

n=0

!(0;2;4)n dH�(!n)cE[ff�ngffng

T ]dH(!n)cei!n�

=

N!�1X

n=0

!(0;2;4)n dH�(!n)c[Gff(!n)]dH(!n)ce

i!n�

(6.57)

where appropriate quantities from equations (6.49) are employed to evaluate the single-sided,

stationary modal force spectral matrix�Gff (!n)

�for circular frequency!n:

�Gff (!n)

�= [�]Tf��ngp

�

npnf�ngT [�] acoustic pressure

�Gff (!n)

�= [�]Tff 0�n gff

0

ngT [�] concentrated forces

�Gff (!n)

�= [�]

Tf�ngu

�

gnugnf�ngT[�] base acceleration

(6.58)

Again, the responses in physical coordinates are obtained from the modal expansion

�[CAA(� )];

�C _A _A(� )

�;�C �A �A(� )

��=

N!�1X

n=0

!(0;2;4)n [�]dH�(!n)c

�Gff (!n)

�dH(!n)c[�]

Tei!n�

(6.59)

These expressions for the modal and physical coordinate response correlation matrices are the

discrete analogs to those given in equations (6.32) and (6.38). The modal/physical response

mean-square values (variances) are determined from the diagonal elements of the corresponding

correlation matrices in the above equations with� = 0.

138

In general, the full expressions in equation (6.57)/(6.59) must be used for calculation of

the modal/physical responses. However, significant computational savings can be realized

by recognizing a simplification for an excitation class. Note that for the cases of a single

concentrated force or multiple contemporaneous forces of the same amplitude, the expression

for the single-sided, stationary spectral matrix can be simplified:

�Gff(!n)

�= [�]T f�gf 0�

n f0

nf�gT [�] (6.60)

So, the general form for the spectral matrix for acoustic pressure, single or multiple equivalent

concentrated forces, or base acceleration can be written as

�Gff (!n)

�= [�]Tf��ngGn

��f�ngT [�] (6.61)

Substituting into equation (6.59) for the physical coordinate correlation matrices results in

�[CAA(� )];

�C _A _A(� )

�;�C �A �A(� )

��=

N!�1X

n=0

!(0;2;4)n [�]dH�(!n)c[�]T f��ngGn

��

f�ngT [�]dH(!n)c[�]T ei!n�

=

N!�1X

n=0

!(0;2;4)n fH�

A(!n)gfHA(!n)gT ei!n�

(6.62)

where

fH�

A(!n)g = [�]dH�(!n)c[�]T f��nggn� =

p2 (6.63)

is the conjugated physical frequency response vector to the excitation amplitude determined by

gn�=p

2Gn��

(6.64)

at any discrete frequency!n.

It is clear that calculation of these physical response correlation matrices has been reduced

to calculation of a frequency response vector. Mean-square (� = 0) physical responses are

determined from

��2q

=

N!�1X

n=0

!(0;2;4)n jfHA(!n)gj2 (6.65)

139

pi(t)

ug(t). . ug(t). .

f'(t)

Figure 6.2 Schematic of simply-supported beam subject to planaracoustic, concentrated force, and base acceleration excitation.

Any excitation that can be put into the form given in equation (6.61) can take advantage of this

efficient means of response calculation. Examples of excitations not satisfying this requirement

include temporally random planar acoustic waves withrandom incidence, turbulent boundary

layer loading, multiple non-contemporaneous concentrated forces, etc.

In the previous sections, forms for the time-dependent load vectors in equation (6.1)

were derived and a modal approach for continuous and discrete frequency domain analysis

of the vibration response of panel-type structures subjected to those loads was developed.

The formulation and implementation will now be validated through comparison with classical

solutions and experimental measurements in the next two sections.


In this section, comparisons will be made with classical analysis for the cases of a simply-

supported aluminum beam and a simply-supported aluminum panel to benchmark the analysis.

All three dynamic, mechanical loads are considered; acoustic pressure, inertial, and concentrated

force. Both the continuous and discrete frequency domain approaches are employed.

6.5.1 Simply-Supported Aluminum Beam

Consider a simply-supported aluminum beam shown schematically in Figure 6.2 with

dimensions of 0.4572x0.0254x0.0023 m (18x1x0.090 inches). The material properties for

140

the specimen are given in Table 6.1. It is assumed that the various loads are mutually exclusive,

and results will be presented as such. Results for continuous and discrete spectral analysis from

the present formulation will be presented. A classical solution for the random response of this

structure subjected to each of the loads was derived for comparison.

Table 6.1 Material properties for aluminum specimens.




Mass Density 2.767e+3 kg/m3 2.588e-4 lbf s2/in4

Coeff. of Thermal Exp. 22.3��/�C 12.4��/�F

The natural frequencies from the classical solution are governed by the equation

2�fn = !n = (n�)2r

EI

ma4(6.66)

wherem is the mass per unit length anda is the length of the beam. The beam was modeled by

a finite element mesh of 36x2 elements. A comparison of the natural frequencies as predicted

by the classical analysis and the finite element analysis are shown in Table 6.2. Excellent

agreement between the two solutions is apparent.

Table 6.2 Comparison of aluminum 0.4572x0.0254x0.0023 m (18x1x0.090 inch)simply-supported beam modal frequencies as predicted by classical and finite element analyses.

Mode Number Classical, Hz FEM, Hz

1 24.76 24.76

2 99.04 99.11

3 222.84 223.20

4 396.15 397.20

The three excitation types are each modeled by white noise with spectral amplitudes which

give consistent loading. For example, a 1g RMS loading over a 10–500 Hz bandwidth is

141

Table 6.3 Comparison of mid–span RMS responses for an aluminum 0.4572x0.0254x0.0023 m(18x1x0.090 inch) simply-supported beam as predicted by classical and finite element analyses.

Analysis RMS Displ. (in) RMS Vel. (in/s) RMS Accel.

m in m/s in/s m/s2 in/s2

Base Acceleration

Classical 7.26e-4 0.0286 0.1138 4.482 — —

FE Continuous 7.21e-4 0.0284 0.1135 4.469 — —

FE Discrete 7.21e-4 0.0284 0.1135 4.469 26.52 1044

Acoustic Pressure

Classical 7.26e-4 0.0286 0.1138 4.482 — —

FE Continuous 7.21e-4 0.0284 0.1135 4.469 — —

FE Discrete 7.21e-4 0.0284 0.1135 4.469 26.52 1044

Concentrated Force

Classical 5.74e-4 0.0226 0.1068 4.205 — —

FE Continuous 5.69e-4 0.0224 0.1063 4.184 — —

FE Discrete 5.69e-4 0.0224 0.1063 4.184 83.64 3293

considered to be a representative inertial loading. This overall load corresponds to a spectral

level of 0.1966 (m/s2)2/Hz (304.704 (in/s2)2/Hz). One can arrive at an equivalent surface pressure

excitation by equating the product of the acceleration level and mass of the beam, which results

in a load of 0.721 N (0.162 lbf), with the product of the unknown surface pressure level and the

surface area of the beam. The resulting equivalent acoustic pressure excitation has an overall

value of 62.1 Pa (0.009 psi) RMS or a spectral level of 7.859 Pa2/Hz (1.65307e-7 (psi)2/Hz) over

a 10–500 Hz bandwidth. Similarly, a concentrated force of 0.721 N (0.162 lbf) RMS or a spectral

level of 1.0596e–3 N2/Hz (5.35595e–5 lbf2/Hz) over 10–500 Hz gives a consistent load when

applied to the beam at mid-span. To make things more interesting, however, the concentrated

force was applied at node 71 of 111 nodes with coordinates of (15, 0.5) inches. Comparisons of

the classical and finite element predictions of the RMS mid-span responses are shown in Table

6.3. A value of 0.02 was used in all cases for the modal critical damping ratio to desensitize the

solutions to frequency resolution. The classical and FE continuous spectral analysis predictions

142

were obtained using the assumption of white noise excitation with a summation over the first four

modes. FE discrete spectral analysis predictions were obtained with a 10–500 Hz band-limited

white noise model, a 0.5 Hz frequency resolution, and summation over the first four modes. The

classical and finite element solutions are in excellent agreement in all cases.

6.5.2 Simply-Supported Aluminum Panel

Consider a simply-supported aluminum panel shown schematically in Figure 6.3 with

dimensions of 0.3556x0.254x0.001 m (14x10x0.040 inches). The material properties for the

specimen are the same as those given in Table 6.1 for the beam case. Again, it is assumed that

the various loads are mutually exclusive. A classical solution for the random response of this

structure subjected to each of the loads was derived for comparison.

f'(t)

ug(t). .

pi(t)

Figure 6.3 Schematic of simply-supported panel subjected to planaracoustic, concentrated force, and base acceleration excitation.

The natural frequencies from the classical solution are governed by the equation

2�fmn = !n = �2

sD

�

��ma

�2+

�nb

�2�(6.67)

143

where� is the mass per unit area,a andb are the panel length and width, andD is the flexural

rigidity given by

D =Eh3

12(1 � �2)(6.68)

The panel was modeled by a finite element mesh of 14x10 elements. A comparison of the natural

frequencies as predicted by the classical analysis and the finite element analysis are shown in

Table 6.4. Negligible discrepancy exists between the two solutions.

Table 6.4 Comparison of aluminum 0.3556x0.254x0.001 m (14x10x0.040 inch)simply-supported panel modal frequencies as predicted by classical and finite element analyses.

Mode Number x-index y-index Classical, Hz FEM, Hz

1 1 1 56.4 56.4

2 2 1 113.7 113.7

3 1 2 168.6 168.6

4 3 1 209.0 209.0

5 2 2 225.8 225.8

6 3 2 321.1 321.2

7 4 1 342.5 342.6

8 1 3 355.4 355.6

9 2 3 412.7 412.8

10 4 2 454.6 454.8

The inertial excitation level used in the previous example will also be used in this case, i.e.,

1g RMS or 0.1966 (m/s2)2/Hz (304.704 (in/s2)2/Hz) over a 10–500 Hz bandwidth. The overall

load of this inertial excitation is 2.49 N (0.56 lbf). The equivalent surface pressure excitation

for this case is 27.58 Pa (0.004 psi) RMS or a spectral level of 1.5524 Pa2/Hz (3.265327e-8

(psi)2/Hz) over a 10–500 Hz bandwidth. Finally, a concentrated force of 2.49 N (0.56 lbf) RMS

or a spectral level of 12.66e–3 N2/Hz (6.4000409e–4 lbf2/Hz) over 10–500 Hz is applied at node

116 of 165 nodes with coordinates of x=0.254, y=0.178 m (x=10, y=7 inches). Comparisons of

the classical and finite element predictions of RMS center-point responses are shown in Table

144

6.5. A value of 0.02 was used in all cases for the modal critical damping ratio. The classical

and FE continuous spectral analysis predictions were obtained using the assumption of band-

unlimited white noise with a summation over the first ten modes. FE discrete spectral analysis

predictions were obtained with a 10–500 Hz band-limited white noise model, a 0.5 Hz frequency

resolution, and summation over the first ten modes. It is clear that excellent comparisons are

achieved in all cases.

Table 6.5 Comparison of center-point RMS responses for an aluminum 0.3556x0.254x0.001 m(14x10x0.040 inch) simply-supported panel as predicted by classical and finite element analyses.

Analysis RMS Displ. RMS Vel. RMS Accel.

m in m/s in/s m/s2 in/ss

Base Acceleration

Classical 2.692e-3 0.0106 0.098 3.845 — —

FE Continuous 2.692e-3 0.0106 0.098 3.845 — —

FE Discrete 2.692e-3 0.0106 0.098 3.844 49.96 1967

Acoustic Pressure

Classical 2.692e-3 0.0106 0.098 3.845 — —

FE Continuous 2.692e-3 0.0106 0.098 3.845 — —

FE Discrete 2.692e-3 0.0106 0.098 3.844 49.96 1967

Concentrated Force

Classical 4.216e-4 0.0166 0.157 6.167 — —

FE Continuous 4.216e-4 0.0166 0.157 6.167 — —

FE Discrete 4.216e-4 0.0166 0.157 6.167 92.63 3647

Comparisons of the displacement power spectral densities PSDs at the coordinates x=0.102,

y= 0.076 m (x=4, y=3 inches) predicted by the classical and finite element analyses for the three

load cases above are presented in Figures 6.4–6.6. In all cases, the classical and finite element

PSDs are indistinguishable. Note the presence of the non-symmetric modes in the concentrated

force response case. PSDs for a random acoustic pressure case with angles of incidence (�i=60�,

�i=45�) are shown in Figure 6.7. This case further demonstrates the excitation of the non-

symmetric modes and will be useful for reference in Chapter 7.

145

Frequency, Hz

Dis

plac

emen

tPS

Din

2 /Hz,

∆f=

0.5

Hz

0 100 200 300 400 50010-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

ClassicalFE ContinuousFE Discrete

Figure 6.4 Classical and finite element displacement response PSDs at x=0.102,y= 0.076 m (x=4, y=3 inches) versus frequency for inertial loading case.

146

Frequency, Hz

Dis

plac

emen

tPS

Din

2 /Hz,

∆f=

0.5

Hz

0 100 200 300 400 50010-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4


Figure 6.5 Classical and finite element displacement response PSDs at x=0.102, y= 0.076m (x=4, y=3 inches) versus frequency for normal acoustic pressure loading case.

147

Frequency, Hz

Dis

plac

emen

tPS

Din

2 /Hz,

∆f=

0.5

Hz

0 100 200 300 400 50010-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4


Figure 6.6 Classical and finite element displacement response PSDs at x=0.102, y=0.076 m (x=4, y=3 inches) versus frequency for concentrated force loading case.

148

Frequency, Hz

Dis

plac

emen

tPS

Din

2 /Hz,

∆f=

0.5

Hz

0 100 200 300 400 50010-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

ClassicalFE DiscreteFE Efficient

Figure 6.7 Classical and finite element displacement response PSDs at x=0.102, y= 0.076 m(x=4, y=3 inches) versus frequency for�i=60�, �i=45� acoustic pressure loading case.

149

It is physically intuitive as to which mechanical loads will excite the non-symmetric modes.

However, some observations about the numerical modeling of such loads is worth mentioning

here. In the discussion that follows, attention is limited to the vector and matrix components

associated with the out-of-plane degrees of freedom only as all other components of the flexural

eigenvectors are zero.

In general, a non-symmetric mode will not be excited by any excitation for which the

corresponding mode shape is orthogonal with respect to the excitation spectral density matrix:

f�gTi[G(!)]f�g

i= 0 (6.69)

Normal acoustic pressure and inertial loading are special cases of this situation as they result in

a real, uniform (full with all entries identical) excitation spectral density matrix. Obviously, the

product of any antisymmetric modal vector transpose and a uniform matrix is the null vector

transpose. The resulting modal force spectral density matrix is sparse with nonzero entries only

at indices corresponding to the symmetric modes.

A concentrated force excitation is modeled by a null spectral density matrix with the

exception of a single nonzero entry located by the index of the degree of freedom at the point

of force application. The product of any modal vector with this matrix is nonzero except for

the case when the nonzero matrix index corresponds to a zero in the modal vector. Hence, all

modes are excited except those for which the point of force application falls on a nodal line.

In this case, the modal force spectral density matrix is real and full, only the modes with nodal

lines containing the point of force application are omitted.

Obliquely incident acoustic pressure excitation gives a complex, full but non-uniform spectral

density matrix. The product of any modal vector with this matrix is nonzero except for cases

when the angles of incidence direct the propagating pressure along an axis of the structure. In

this special case, portions of the excitation spectral density matrix are uniform and these portions

150

have indices corresponding to the antisymmetric nature of even-odd modes (e.g.,�i=60�, �i=0�

does not excite the *,2 modes). So, with the exception of the modes excluded due to the latter

situation, the modal force spectral density matrix is complex and full. One final note should be

made regarding the extent to which the antisymmetric modes are excited. Modes occurring at

lower frequencies are excited more weakly by excitations of this type because the wavelength

of the propagating pressure may be long relative to the extent of the structure. This causes the

incident pressure to appear to be close to uniform. This is exemplified in the case presented in

Figure 6.7, where the pressure wavelengths range from 34.29 to 0.686 m (112.5 to 2.25 feet)

at 10 Hz and 500 Hz, respectively. In this case, all antisymmetric modes are excited weakly

as is evident in the figure.

The finite element analysis has been shown to produce very accurate results in comparison

with classical analysis for all three loading types for simply-supported beam and panel structures.

Now, attention will be turned to experimental configurations for validation of the analytical tool

with experimental measurement.


Comparisons will be made with experimental measurements for the cases of a clamped-

clamped aluminum beam, a clamped-clamped glass-epoxy beam to assess model accuracy incre-

mentally, prior to making comparisons with measurements from a clamped-clamped SMAHC

beam. Results will be shown for the SMAHC beam at several temperatures indicative of various

states in the beam thermomechanical response. Finally, comparisons will be made for the case

of a fully-clamped aluminum panel as an introduction to the next chapter.

All of the experimental beam vibration results presented in this study result from base

acceleration (inertial) excitation. A picture of the overall apparatus is shown in Figure 6.8.

The shaker system consists of a large electromechanical shaker configured horizontally and

151

attached to a magnesium slip table, which oscillates in a horizontal plane during operation. The

specimen is mounted in the fixture with a vertical orientation to the width-direction in order to

avoid complications due to gravity. The excitation acceleration is measured by accelerometers

(PCB model T352M92) placed at each end of the specimen fixture. The normal velocity and

acceleration responses of the beam are measured with a laser Doppler velocimeter (Ometron,

VPI) and accelerometers (Endevco model 2250A-10). It is obvious that the beam response

transducers measure the absolute motion of the beam specimens (including the base motion).

Most structural dynamic prediction tools, in fact all of those used in this study, are formulated

to provide relative motion response for such configurations. Therefore, it is imperative that

experimental procedures allow for the removal of the base motion contributions. The importance

is not so much for comparison of response amplitude as the base motion is typically much less

than that of the flexible structure, but for comparison of frequency response or response spectra.

Shaker

Slip Table

IR Camera

LaserVibrometer

Beam/FixtureAssembly

Figure 6.8 Overall configuration for base acceleration tests on beam specimens.

To allow the most flexibility in the procedure of base motion removal, a redundant accelerom-

eter was attached to each end of the support fixture and connected to signal conditioning with

hardware integration. The redundant reference accelerometer signals were integrated once to

result in velocity measurements for subtraction from the laser vibrometer output. Subsequently,

152

the base motion can be removed from the acceleration or velocity responses in either the time

domain, by simple subtraction, or in the frequency domain as follows. This discussion will

focus on the accelerometer measurements, but it equally valid for any response quantity. The

relative acceleration can be written as

a = as � af (6.70)

where the specimen and fixture accelerations have been denoted byas and af . For stationary

stochastic response, the covariance can be obtained from the following expectation

E[a(t)a(t� � )] = E��as(t)� af(t)

��as(t� �)� af (t� � )

��(6.71)

This can be written in the frequency domain as

�aa(!) = �asas(!)� �asaf (!)� �afas(!) + �afaf (!) (6.72)

where the� are power and cross spectral density functions. Thus, it is clear that frequency

domain measurements (i.e., spectral matrix) can be post-processed to obtain the relative motion

spectra. In this study, the time data was collected and post-processed to remove the base motion

and generate relative response spectra.

6.6.1 Clamped Aluminum Beam

A picture of the apparatus for the clamped-clamped aluminum beam configuration is shown

in Figure 6.9. The fixture is the same as that described in Chapter 5, but the electrical connection

blocks and electrical/thermal insulation layers have been removed. The overall beam dimensions

are 0.5588x0.0254x0.0023 m (22x1x0.09 inches) with an unsupported length of 0.4572 m (18

inches). The overall beam mass was measured to be 89.79e-3 kg (0.198 lbf). The elastic

properties of the material were also measured and are given in Table 6.6.

The beam was excited by random base acceleration over a 20–400 Hz bandwidth with a

RMS value of 0.25 g. The beam response was measured by the laser vibrometer and two

153

Fixture

Response AccelsIntegrated Accel

Control Accel

12

Figure 6.9 Aluminum beam specimen mounted in base acceleration test configuration.

Table 6.6 Material properties for aluminum beam specimen in dynamic response test.

Young’s Modulus 74.47 GPa 10.8 Msi



Density 2.767e+3 kg/m3 2.588e-4 lbf•s/in4

Mode Number Modal Critical Damping Ratio

1 0.0025

2 0.001

3 0.00086

accelerometers at locations 0.10 m (4 inches) and 0.25 m (10 inches) from right grip, indicated

by position numbers 1 and 2 in Figure 6.9. Time records with an overall length of 200 seconds

were collected with a sampling rate of 1024 Hz to allow 50–frame averages with a bandwidth

of 0–400 Hz and a frequency resolution of 0.25 Hz. The time data were post-processed to

construct relative response quantities, frequency response functions between the beam response

transducers and the fixture transducers, and power spectral densities (PSDs) of the acceleration

input and relative beam responses.

The beam was modeled with a 36x2 mesh of the 24 degree-of-freedom rectangular plate

elements described in Chapter 4. Modal damping estimates from measured frequency response

functions, see Table 6.6, were used in the predictions. The beam grips were torqued to 13.56

154

N•m (120 in•lbf), which gave a measured fundamental frequency of 55.5 Hz. The fundamental

frequency with perfectly clamped boundaries was predicted to be 58.4 Hz. The boundary

conditions were modeled as clamped with the exception of torsional springs, with a spring

rate of 113 N•m/rad (1000 in•lbf/rad) applied to the longitudinal-slope degrees of freedom to

match the fundamental frequency. Only two symmetric modes were expected in the excitation

bandwidth, but a total of 10 modes was used in the predictions to account for the residual effects

of the higher order modes. A uniform modal critical damping ratio of 0.1% was used for modes

4–10. The influence of the accelerometers was modeled as lumped masses (�0.4e-3 kg, 2.283e-6

lbf•s2/in4) at appropriate nodes. The input acceleration PSDs, as measured by one of the control

accelerometers, was taken as input for the predictions.

Comparisons of predicted and measured displacement PSDs for measurement locations 1

and 2 are shown in Figures 6.10 and 6.11. Note that the “measured” displacement PSDs were

generated by integrating the acceleration responses in the frequency domain. It can be seen that

excellent agreement is achieved, except in overall RMS response level. The discrepancy in RMS

level is due to the fact that the damping is so light in this case that the frequency resolution of

0.25 Hz is insufficient to adequately capture the peak values in the FRFs or spectra. Of course,

the spectral analysis conserves the total response energy by spill-over into adjacent frequency

bins. This spill-over results in slightly broader response peaks with lower amplitude. Curve

fitting techniques for estimation of modal parameters misinterpret this as higher damping values.

The main culprit in this case is the damping value at the fundamental frequency, where the

response peak is extremely sharp. A damping value of approximately 0.1% would probably be

more representative of the actual damping value in the first mode because it is consistent with

that at mode 3. With this change in damping, the displacement response RMS values at the two

measurement locations are 1.24e-4 m and 2.54e-4 m, respectively, in nearly perfect agreement

with measurement.

155

Frequency, Hz

Dis

pla

cem

entP

SD

,m2 /H

z

100 200 300 40010-20

10-18

10-16

10-14

10-12

10-10

10-8

10-6

Measurement, Pos 1, 1.24e-4 mRMSPrediction, Pos 1, 0.84e-4 mRMS

Figure 6.10 Measured and predicted displacement PSDcomparison for position 1 on the aluminum beam (�f=0.25 Hz).

156

Frequency, Hz

Dis

pla

cem

entP

SD

,m2 /H

z

100 200 300 40010-20

10-18

10-16

10-14

10-12

10-10

10-8

10-6


Figure 6.11 Measured and predicted displacement PSDcomparison for position 2 on the aluminum beam (�f=0.25 Hz).

157

Table 6.7 Material properties for glass–epoxy beam specimen in dynamic response test.

E1 49.30 GPa 7.15 Msi

E2 20.00 GPa 2.90 Msi

G12 9.65 GPa 1.40 Msi

�12 0.29 0.29

� 2.031e+3 kg/m3 0.19e-3 lbf•s/in4

Mode Number Modal Critical Damping Ratio

1 0.0055

2 0.005

3 0.0048

4 0.005

6.6.2 Clamped Glass-Epoxy Beam

The performance of the FE analysis will now be tested against a baseline composite beam.

A glass-epoxy beam with dimensions of 0.5588x0.0254x0.0018 m (22x1x0.072 inches) with an

unsupported length of 0.4572 m (18 inches) was tested in the same experimental configuration as

that for the aluminum beam. The overall beam mass was measured to be 53.9e-3 kg (0.119 lbf).

The elastic properties of the material system were described in Chapter 3. The material properties

at ambient temperature were used to generate the predicted responses. Those properties are

reiterated in Table 6.7 for convenience.

The beam excitation and data acquisition configuration was the same as that for the aluminum

beam with the exception of the excitation bandwidth, which was changed to 10–400 Hz because

of the lower fundamental frequency in this case. The overall excitation level of 0.25 gRMS was

maintained. The measurement locations will again be referred to as positions 1 and 2. Time data

were collected and post-processed to construct relative response quantities, frequency response

functions between the beam response transducers and the fixture transducers, and power spectral

densities (PSDs) of the acceleration input and relative beam responses.

158

An analogous FE model was also developed for this case. Modal damping values were

estimated from the measured frequency response functions, see Table 6.7, and used in the

predictions. In this case, a torque of 13.56 N•m (120 in•lbf) on the beam grips resulted in

a measured fundamental frequency of 33.5 Hz. The fundamental frequency with perfectly

clamped boundaries was predicted to be 35 Hz. Torsional springs with a rate of 19.21 N•m/rad

(170 in•lbf/rad) were applied to match the fundamental frequency. Again, the input acceleration

PSDs, as measured by one of the control accelerometers, were taken as input for the predictions.

Predicted and measured displacement PSDs for measurement locations 1 and 2 are shown

in Figures 6.12 and 6.13. Again, excellent agreement is achieved, including the RMS response

levels. There is a slight over-prediction of the second symmetric modal frequency, but the

comparison is within experimental uncertainty.

The dynamic analysis has been validated by comparison with the measured response of

conventional structures. Attention will now turn to demonstration of the dynamic response

performance of a SMAHC beam specimen and validation of the formulation by comparison of

the measured and predicted responses.

6.6.3 Clamped SMAHC Beam

The SMAHC beam specimen mounted in the support fixture and configured for base

acceleration testing is shown in Figure 6.14. The configuration is essentially the same as

that described in the Experimental Validation section of Chapter 5. The beam is supported

by the mechanical grips indicated in the figure by “m”. The grips designated as “e” are

electrical connections for the Nitinol leads, but also provide mechanical restraint. The electrical

connections are isolated from the fixture by a layer of fibrous ceramic insulation. The beam is also

thermally isolated from the fixture by a layer of this insulation on both sides. The beam is heated

by DC electrical current controlled by a thermocouple measurement located approximately on

the beam centerline and 0.0127 m (0.5 in) from the right mechanical grip, indicated in the figure

159

Frequency, Hz

Dis

pla

cem

entP

SD

,m2 /H

z

100 200 300 40010-20

10-18

10-16

10-14

10-12

10-10

10-8

10-6


Figure 6.12 Measured and predicted displacement PSD comparisonfor position 1 on the glass-epoxy beam (�f=0.25 Hz).

160

Frequency, Hz

Dis

pla

cem

entP

SD

,m2 /H

z

100 200 300 40010-20

10-18

10-16

10-14

10-12

10-10

10-8

10-6


Figure 6.13 Measured and predicted displacement PSD comparisonfor position 2 on the glass-epoxy beam (�f=0.25 Hz).

161

m e

SMAHC Beam

Fixture

Power Cables

tc

tc

tc

Figure 6.14 Front view of the SMAHC beam specimenmounted in clamping fixture for dynamic testing.

by “tc”. The thermal controller consists of a process controller (Omega model CN77353–PV),

and a DC power supply (HP model 6652A). Thermocouples also monitor the temperature at

each end of the fixture, indicated in Figure 6.14, and one measures the ambient air temperature.

A rear view of the beam/fixture assembly is given in Figure 6.15, which shows additional

detail on the dynamic experimental setup. The placement of the control thermocouple is more

apparent in this view as is the presence of the electrical/thermal insulation. The base acceleration

excitation was controlled with the average of the two control accelerometers. The “integrated

accelerometers” were utilized in the calculation of relative velocity response from the laser

vibrometer data, as described previously. The positions of the two response accelerometers

are consistent with the descriptions of the previous tests and are again designated by position

numbers 1 and 2.

The beam was excited by random base acceleration over a 10–400 Hz bandwidth with a

RMS value of 0.25 g. Time records with an overall length of 200 seconds were collected with

a sampling rate of 1024 Hz to allow 50–frame averages with a bandwidth of 0–400 Hz and

a frequency resolution of 0.25 Hz. The time data were post-processed to construct relative

response quantities, frequency response functions between the beam response transducers and

162

Fixture

Control TC Response Accels

Control Accel

IntegratedAccel

12

Figure 6.15 Rear view of the SMAHC beam specimenmounted in clamping fixture for dynamic testing.

the fixture transducers, and power spectral densities (PSDs) of the acceleration input and relative

beam responses. Dynamic data were collected at ambient and at elevated temperatures from

32.22�C (90�F) to 121.11�C (250�F) in 5.56�C (10�F) increments. The beam mechanical and

ribbon electrical grips were torqued to 13.56 N•m (120 in•lbf) just prior to dynamic excitation

to minimize effects of changing environmental conditions. The fixture temperature increased by

approximately 2.22�C (4�F) during the course of data collection, which was in almost direct

proportion to the increase in ambient air temperature.

First, some general observations will be made concerning the experimental behavior and

results. Because of the material system characteristics and test configuration, the beam exhibited

a thermal post-buckling deflection over range of temperatures, as described in detail in Chapter

5. The beam exhibited some interesting dynamic behavior (such as intermittent snap-through)

within this temperature range, particularly at the point of loss of stability in the flat configuration

and then at the point of regaining stability in the flat configuration, as expected. No attempt will

be made to predict the dynamic behavior within this range, although it is possible with the present

analysis for the portions where the dynamic response is geometrically linear about a nonlinear

static deflection. Although the beam was returned to a flat configuration at approximately

54.44�C (130�F), the presented results will focus on higher temperatures. The forced vibration

amplitude was visually noticeable in the inactive state (ambient temperature), but was virtually

163

Table 6.8 Summary of RMS measured displacements at position 2on the SMAHC beam and modal parameters versus temperature.

Temp.,�C RMS Displ., m f1, Hz �1, % f3, Hz �3, %

23 0.322e-3 26.4 0.89 160.3 0.95

— — — — — —

60 0.15e-3 83.1 0.10 279.3 0.45

65.56 0.11e-3 94.4 0.13 309.4

71.11 0.05e-3 107.0 0.50 342.7 0.16

76.67 0.063e-3 118.3 0.19 372.5 0.28

82.22 0.068e-3 126.7 0.15 — —

87.78 0.063e-3 130.9 0.14 — —

93.33 0.062e-3 134.0 0.14 — —

98.89 0.066e-3 136.0 0.11 — —

104.4 0.067e-3 137.6 0.11 — —

110.0 0.073e-3 139.2 0.10 — —

115.6 0.072e-3 138.7 0.10 — —

121.1 0.067e-3 140.0 0.09 — —

undetectable at temperatures above 71.11�C (160�F). No evidence of delamination or other flaws

were detected subsequent to testing, despite the large thermal post-buckling deflection and rather

high temperature developed during the tests.

Other observations can be made from a summary of RMS displacement responses and modal

parameter estimates shown as a function of temperature in Table 6.8. Temperatures for which the

beam was buckled are excluded from the table. The RMS displacements correspond to location

2 and were calculated in the frequency domain from the corresponding accelerometer PSDs. It

can be seen that, although a majority of the stiffening effect was achieved at approximately

93.33�C (200�F), performance remains good far beyond that temperature with a maximum

increase in the fundamental frequency of 430% at 121.11�C (250�F). The variation of the modal

frequencies with temperature is more easily seen in Figure 6.16. There, it can be seen that the

fundamental frequency starts to reach an asymptotic value at approximately 93.33�C (200�F)

164

Temperature, oC

Fre

que

ncy

,Hz

20 30 40 50 60 70 80 90 100 110 1200

50

100

150

200

250

300

350

400

Mode 1

Mode 3

Figure 6.16 Measured mode 1 and mode 3 naturalfrequencies of the SMAHC beam versus temperature.

165

and mode three escapes the excitation bandwidth at temperatures above 82.22�C (180�F). Note

the peculiar changes in the mode–1 damping near 71.11�C (160�F). This effect is reflected in the

measured displacement response RMS values and can be seen in Figure 6.17, which shows the

measured displacement response PSD at ambient and three elevated temperatures. The four PSDs

correspond to the filled symbols in Figure 6.16. It can be seen that the maximum attenuation in

the RMS displacement response is achieved at 71.11�C (160�F), where there is maximum peak

response reduction due to higher damping, although the stiffening effect continues to increase at

higher temperatures. One possible explanation is that there may be an optimal state near 71.11�C

(160�F) where there is adequate austenite and dynamic stress and low enough temperature to

allow enhanced damping by stress induced martensitic (SIM) transformation hysteresis. At

higher temperatures, the dynamic stresses are insufficient to drive the SIM transformation as it

becomes increasingly difficult to induce.

Recall that the beam dimensions are 0.5588x0.0254x0.0019 m (22x1x0.078 in) with a

lamination of (45/0sma/-45/90)2s. Five ribbon widths are embedded within each 0� layer. Each

ribbon has a cross section of 2.29e-3x1.52e-4 m (0.09x0.006 inches). This configuration was

modeled as a Nitinol volume fraction of 0.5538 in the 0� layers, for an overall volume fraction

of 13.8%. The mass densities of the glass-epoxy and Nitinol ribbon (prestrained 4%) were

found to be 0.19e-3 lbf•s2/in4 and 0.5349e-3 lbf•s2/in4. With the beam/ribbon dimensions, the

trimmed ribbon length of 0.66 m (26 inches) and the measured mass densities, the calculated

mass of the beam is 75.5e-3 kg (0.166 lbf). This is within 1% of the measured overall mass

of the SMAHC beam, which was found to be 74.92e-3 kg (0.165 lbf). The thermomechanical

properties of the constituent materials were described in Chapter 3 and are given in Tables 3.5

and 3.8. Recall that the composite cure cycle constitutes thermal cycle number one (1) on the

Nitinol ribbon. This was the first test of the SMAHC beam specimen, which corresponds to

166

Frequency, Hz

Dis

pla

cem

entP

SD

,m2 /H

z

100 200 300 40010-18

10-16

10-14

10-12

10-10

10-8

10-6

Measurement, 74oFMeasurement, 160oFMeasurement, 200oFMeasurement, 250oF

Figure 6.17 Measured displacement PSDs for position 2 on theSMAHC beam at ambient and elevated temperatures (�f=0.25 Hz).

167

thermal cycle number two (2). The appropriate recovery stress data from Table 3.8 were used

in the model to predict the dynamic response.

The element properties for a 32x2-element-mesh model of this beam were evaluated as a

function of temperature from these material properties. Modal damping estimates were taken

from the experimental data in Table 6.8. A total of ten modes was used in the predictions in all

cases to account for residual effects. The damping was assumed to be 0.1% on all antisymmetric

modes and modes out of bandwidth. In this case, the torque of 13.56 N•m (120 in•lbf) on the

beam grips resulted in a measured fundamental frequency of 26.4 Hz at ambient temperature.

The fundamental frequency with perfectly clamped boundaries was predicted to be 37.6 Hz.

The boundary conditions in this test are not as close to clamped conditions as in the previous

tests because of the insertion of the thermal insulating material between the beam and grip

surfaces. Torsional springs with a rate of 1.98 N•m/rad (17.5 in•lbf/rad) were applied to match

the fundamental frequency. The influence of the accelerometers was modeled as lumped masses

(˜0.4e-3 kg, 2.283e-6 lbf•s2/in4) at appropriate nodes. Again, the input acceleration PSDs, as

measured by one of the control accelerometers, was taken as input for the predictions.

Measured and predicted displacement PSDs for measurement location 2 on the SMAHC

beam at four temperatures are shown in Figures 6.18–6.21. The figures show results for

ambient temperature, 71.11�C (160�F), 93.33�C (200�F) and 121.1�C (250�F), respectively,

and are representative of the comparisons at all temperatures. The correlation at ambient

temperature is excellent with less than 9% difference between measured (3.23 mRMS, 0.0127

inRMS) and predicted (2.96 mRMS, 0.0116 inRMS) RMS response level. The comparisons at

elevated temperature show some discrepancy, but are in excellent qualitative agreement. The

predicted responses consistently show slightly higher modal frequencies and lower RMS levels

than the corresponding experimental measurements. This trend is due to an over-prediction of

168

Frequency, Hz

Dis

pla

cem

entP

SD

,m2 /H

z

100 200 300 40010-18

10-16

10-14

10-12

10-10

10-8

10-6

Measurement, 23.3oCPrediction, 23.3oC

Figure 6.18 Measured and predicted displacement PSD comparison forposition 2 on the SMAHC beam at ambient temperature (�f=0.25 Hz).

169

Frequency, Hz

Dis

pla

cem

entP

SD

,m2 /H

z

100 200 300 40010-18

10-16

10-14

10-12

10-10

10-8

10-6


Figure 6.19 Measured and predicted displacement PSD comparisonfor position 2 on the SMAHC beam at 71.1�C (160�F, �f=0.25 Hz).

170

Frequency, Hz

Dis

pla

cem

entP

SD

,m2 /H

z

100 200 300 40010-18

10-16

10-14

10-12

10-10

10-8

10-6


Figure 6.20 Measured and predicted displacement PSD comparisonfor position 2 on the SMAHC beam at 93.3�C (200�F, �f=0.25 Hz).

171

Frequency, Hz

Dis

pla

cem

entP

SD

,m2 /H

z

100 200 300 40010-18

10-16

10-14

10-12

10-10

10-8

10-6


Figure 6.21 Measured and predicted displacement PSD comparison forposition 2 on the SMAHC beam at 121.1�C (250�F, �f=0.25 Hz).

172

the recovery force stiffening effect and is attributable to boundary condition modeling issues

described in the Experimental Validation section of Chapter 5.

Recall from that discussion that the in-plane boundaries of the physical system are somewhat

compliant due to the layer of insulation between the mechanical and electrical grips. The

boundary compliance partially relieved the recovery stress and resulted in a reduced stiffening

effect. This effect was captured in the model by applying axial springs with a spring rate of

1.97e+6 N/m (11,250 lbf/in) to the in-plane degrees of freedom at the mechanical grips. The

effect of fixture expansion with temperature was also modeled by applying appropriate enforced

displacements at the beam ends. New predicted responses were made with these changes. The

measured and predicted displacement PSDs for measurement location 2 on the SMAHC beam

are shown in Figures 6.22–6.25. Excellent agreement is achieved between the predicted and

measured responses at all temperatures.

The formulation and solution procedures have now been fully exercised and validated against

a variety of experimental results, most importantly with the static and dynamic response of a

SMAHC beam. Further work was done to validate the acoustic radiation model. Experimental

validation was attempted for the case of a clamped aluminum panel. Dynamic responses of the

panel due to acoustic excitation were measured during the acoustic radiation tests and validation

of the dynamic response prediction capability for this case will be discussed in the next section.

6.6.4 Clamped Aluminum Panel

Experiments were conducted to measure the structural-acoustic response of an aluminum

panel subject to broadband, normal acoustic excitation. The experiments were performed in the

Transmission Loss Apparatus (TLA) at NASA Langley Research Center, shown schematically

in Figure 6.26. The results of the experiments were reported by Turner and Rizzi [55], but are

briefly reiterated here. Vibration results are reported in this chapter and acoustic radiation results

are reported in the Experimental Validation section of Chapter 7.

173

Frequency, Hz

Dis

pla

cem

entP

SD

,m2 /H

z

100 200 300 40010-18

10-16

10-14

10-12

10-10

10-8

10-6


Figure 6.22 Measured and predicted displacement PSD comparison forthe SMAHC beam at ambient temperature with boundary effects (�f=0.25 Hz).

174

Frequency, Hz

Dis

pla

cem

entP

SD

,m2 /H

z

100 200 300 40010-18

10-16

10-14

10-12

10-10

10-8

10-6


Figure 6.23 Measured and predicted displacement PSD comparison forthe SMAHC beam at 71.1�C with boundary effects (160�F, �f=0.25 Hz).

175

Frequency, Hz

Dis

pla

cem

entP

SD

,m2 /H

z

100 200 300 40010-18

10-16

10-14

10-12

10-10

10-8

10-6


Figure 6.24 Measured and predicted displacement PSD comparison forthe SMAHC beam at 93.3�C with boundary effects (200�F, �f=0.25 Hz).

176

Frequency, Hz

Dis

pla

cem

entP

SD

,m2 /H

z

100 200 300 40010-18

10-16

10-14

10-12

10-10

10-8

10-6


Figure 6.25 Measured and predicted displacement PSD comparison for theSMAHC beam at 121.1�C with boundary effects (250�F, �f=0.25 Hz).

177

��

��

��

Scanning laservibrometer

*

R4

R2 R1 S

R6R5

R7

R3 Speaker

Source roomReceiving room

Figure 6.26 Transmission loss apparatus schematic.

The panel was installed in a picture-frame-type clamping fixture such that the unsupported

dimensions were 0.305x0.178x1.02e–3 m (12�7�0.040 inches). The entire assembly was

mounted in the TLA partition baffle and subjected to normal-incidence, acoustic excitation

provided by a speaker positioned approximately 0.914 m (36 inches) from the panel. The

speaker was driven by a random signal, band-limited to a frequency range of 100–500 Hz. A

centered 10�5 inch grid of 11 by 6 locations (total of 66 locations) was defined on the panel

surface for vibration response measurements. Power spectra of the panel normal velocity at these

locations were acquired with the laser vibrometer and a spectrum analyzer.

The incident acoustic pressure was measured at several locations in the plane of the panel,

with the panel and clamping fixture removed from the baffle, under identical source conditions

to that of the response measurements. These source measurements were collected to assess the

uniformity of the incident pressure across the panel and to provide input for the computational

procedure. The overall sound pressure level in the plane of the panel at the opening center was

approximately 109 dB (ref 20�Pa, 2.9E-9 psi).

A 12x7-element mesh of the 24 degree-of-freedom rectangular plate elements was used to

178

model the structure. Torsional boundary springs of 15.82 N•m/rad (140 in•lbf/rad) were employed

to match the fundamental measured frequency. The source pressure spectrum obtained in the

plane of the panel at the center of the baffle hole, when the panel and fixture were removed, was

doubled and used as the blocked pressure input to determine the predicted panel response and

acoustic radiation. The predicted results were generated using the following material properties

for aluminum: E=10 Msi,�=0.33, and�=2.5756e–4 lbf s2/in4.

The power spectral density of the pressure incident upon the panel is shown in Figure 6.27.

Comparisons of the predicted and measured panel normal velocity are shown in Figures 6.28

and 6.29 for two measurement locations; near the panel center and in the lower-right quadrant.

The measurement location coordinates indicated in the figure are relative to the panel’s lower

left corner. The agreement is excellent with the exception of the appearance of additional peaks

in the measured spectra. These peaks are attributable to the first and second non-symmetric

modes of the panel. A number of factors could cause this type of phenomenon. The most likely

culprit is the acoustic loading, but the source measurements indicated that the acoustic load was

very nearly uniform and normal. Furthermore, it is expected that antisymmetric modes would

be excited very weakly by acoustic pressure loading anomalies as previously discussed in the

simply-supported aluminum panel section. It was later found that the panel was prestressed

during fixturing, and it is believed that this is the reason for the significant antisymmetric modal

response.

179

Frequency, Hz

Pre

ssur

eP

SD

,(p

si)2 /H

z

0 200 400 600 80010-13

10-12

10-11

10-10

10-9

10-8

Figure 6.27 PSD of the acoustic excitation incident upon the panel (OASPL 109 dB, ref 20�Pa).

180

Frequency, Hz

Vel

oci

tyP

SD

,(in

/s)2 /H

z

100 200 300 400 50010-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

ExperimentFinite Element

Figure 6.28 Comparison of predicted and measured panel normal velocity PSD atcoordinates of x=0.152, y=0.102 m (x=6, y=4 inches) relative to panel lower left corner.

181

Frequency, Hz

Vel

oci

tyP

SD

,(in

/s)2 /H

z

100 200 300 400 50010-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100


Figure 6.29 Comparison of predicted and measured panel normal velocity PSD atcoordinates of x=0.229, y=0.051 m (x=9, y=2 inches) relative to panel lower left corner.

182

The dynamic load models and spectral analysis method used in this study was presented in

this chapter. The formulation and procedures were validated against known solutions prior

to attempting correlation with experiment. The analysis was incrementally validated with

experimental measurements for the cases of inertially loaded beams including an aluminum

beam, a glass-epoxy beam and a SMAHC beam. Experimental results from the SMAHC beam

specimen show enormous control authority over the resonance frequencies and RMS response

level. Excellent agreement was achieved between prediction and experimental measurement, and

reasons for existing discrepancies were discussed. The chapter is concluded with comparisons

of the measured and predicted responses of a fully-clamped aluminum panel in preparation for

acoustic radiation analysis and validation, which will be presented in the next chapter.

183

Chapter 7Acoustic Radiation

The equations derived in Chapter 4 governing the static and dynamic response of panel-

type SMA hybrid composite structures subjected to thermal and mechanical loads are applicable

to a variety of geometric configurations and arbitrarynatural boundary conditions. However,

acoustic radiation and transmission loss analyses are only practical for panel-type structures

with constraints around the entire perimeter of the structure. Furthermore, it is assumed that the

structure is supported in such a way that flanking paths for the propagation of acoustic energy are

prohibited, i.e., acoustic energy can only be transmitted by the flexible structure. Thus, consider

a rectangular composite panel supported in an infinite rigid baffle (infinitely stiff/massive wall)

separating two semi-infinite acoustic media. This configuration is typically simulated in practice

by a transmission lossfacility, shown schematically in Figure 6.26. Acoustic radiation analysis

will be limited to the cases of acoustic pressure or concentrated force excitation on thesource

side of the panel. The panel radiates acoustic pressures to thereceivingacoustic field due to its

dynamic response. The equations governing acoustic radiation responses under these conditions

will be developed in this chapter using continuous and discrete frequency domain approaches.

Consider a panel supported in an infinite rigid baffle and radiating to a semi-infinite acoustic

field, shown schematically in Figure 7.1. The acoustic pressure radiated by the panel to a point

(x; y; z) in the acoustic field can be related to the dynamic response of the panel through the

184

z η,y

θ

φξ,x

(ξ,η)

rR

(x,y,z)

Figure 7.1 Schematic of a panel radiating to a semi-infinite acoustic field.

use of Rayleigh’s integral [56]:

pt(x; y; z; t) =

Z

A

�

2�r�w��; �; t�

r

c

�dA (7.1)

where� is the fluid density,�w is the out-of-plane acceleration response of the panel,� and� are

spatial coordinates in the plane of the panel,r is the distance from(�; �) on the plate to(x; y; z)

in the acoustic field,c is the acoustic wave speed,t � rc

is the retarded time between the two

points, andA is the surface area of the panel. The subscriptt indicates transmitted pressure to

be consistent with the nomenclature in Chapter 6.

7.1 Continuous Spectral Analysis

The cross-correlation of the radiated pressure can be formed from the following expected

value

Cpp

�~x; t1; ~x

0; t2�= E

�pt(x; y; z; t1)pt

�x0; y0; z0; t2

��

=� �

2�

�2Z

A

Z

A0

1

r1r2Eh�w��; �; t1 �

r1

c

��w��0; �0; t2 �

r2

c

�idA dA0 (7.2)

185

where the expected value expression in the integrand is the cross-correlation function of the

plate normal accelerationC �w �w

��; �; t1 �

r1c; �0; �0; t2 �

r2c

�, r1 is the distance from(�; �) on the

structure to~x = (x; y; z) in the acoustic field, andr2 is the distance from(�0; �0) on the structure

to ~x0 = (x0; y0; z0) in the acoustic field. For stationary response, the time lag can be defined as

� = t2 � t1 so that @2

@t21

@2

@t22

= @4

@�4and the assumption of mean-square differentiability allows

equation (7.2) to be rewritten in the form

Cpp

�x; y; z; x0; y0; z0; � 0

�=� �

2�

�2 Z

A

Z

A0

1

r1r2

@4

@� 4Cww

��; �; �0; �0; � 0

�dA dA0 (7.3)

where� 0 = � + r1�r2c

. Because the plate normal deflection cross-correlation function is related

to the cross spectral density function through the Wiener-Khinchine relation

Cww

��; �; �0; �0; � 0

�=

1Z�1

Sww

��; �; �0; �0; !

�ei!�

0

d! (7.4)

and the response is stationary, the following expression can be derived

@4

@� 4Cww

��; �; �0; �0; � 0

�=

1Z

�1

!4Sww

��; �; �0; �0; !

�ei!�

0

d! (7.5)

Substituting in equation (7.3) results in the following expression for the cross-correlation of the

radiated pressure.

Cpp

�x; y; z; x0; y0; z0; � 0

�=� �

2�

�2 Z

A

Z

A0

1

r1r2

1Z�1

!4Sww

��; �; �0; �0; !

�ei!�

0

d! dA dA0 (7.6)

Recall that the displacement field of the panel is known on the elemental level due to the

finite element spatial discretization, shown in equation (4.5):

we(�; �) � fLw(�; �)gT fabg (7.7)

wherefLw(�; �)g is the vector of out-of-plane interpolation functions andfabg is the out-of-

plane displacement vector, both for elemente. Then, the displacement cross-correlation function

between portions of the plate belonging to elementsi andj can be approximated by

C ijww(�; �) � fLw(�; �)g

Ti E

hfabgifabg

Tj

i�Lw

��0; �0

�j

(7.8)

186

whereE[fabgifabgTj ] = [C

ijaa] is the physical displacement correlation matrix between elements

i and j. Consequently, the panel area integrals in equation (7.6) can be replaced by a double

sum over element area integrals to obtain

Cpp =� �

2�

�2 NeXi=1

NeXj=1

Z

Ai

Z

A0

j

1

r1r2fLw(�; �)g

Ti

1Z�1

!4[Sijaa(!)]e

i!� 0

d!�Lw

�� 0; �0

�jdAi dA

0

j

(7.9)

whereNe is the number of elements. It follows that the mean square(� = 0) radiated pressure

at the point(x; y; z) in the acoustic field has the form

p2t (x; y; z) =� �

2�

�2 NeXi=1

NeXj=1

Z

Ai

Z

A0

j

1

r1r0

1

fLw(�; �)gTi

1Z�1

!4[Sijaa(!)]e

i!r1�r0

1

c d!

�Lw

��0; �0

�jdAi dA

0

j

(7.10)

where

r1 =

q(x� �)

2+ (y � �)

2+ z2 (7.11)

and

r01=

q(x� �0)

2+ (y � �0)

2+ z2 (7.12)

are distances from the same point in the acoustic field to different points on the panel and

(r1 � r01)=c is the zero-time-lag retarded time difference (the difference in the time it takes for

an acoustic disturbance to travel from(�; �) to (x; y; z) versus(�0; �0) to (x; y; z)).

Note that the spectral density matrix of the physical displacement response in the integrand

of the spectral integral in equation (7.10):1Z

�1

!4[Sijaa(!)]e

i!r1�r0

1

c d! (7.13)

is a subset of the system physical displacement spectral density matrix. The form for the system

matrix is inferred from equation (6.38)

[SAA(!)] = [�]dH�(!)c�Sff (!)

�dH(!)c[�]

T (7.14)

187

where [�] is the modal matrix,dH(!)c is a diagonal matrix of stationary frequency response

functions, and the modal force spectral density matrix is given by the appropriate form in

equations (6.34). So, an arbitrary element(m;n) of the spectral integrand in equation (7.10)

requires evaluation of an improper integral of the form

1Z

�1

!4H�

m(w) Hn(!)Smn��

(!)ei!r1�r0

1

c d! (7.15)

where the frequency response functionsHm(!) are given by equation (6.29). Note that these

integrals have the same form as those required for determining modal correlation functions of

the structure’s acceleration response for nonzero time lag(� 6= 0), shown in equation (6.36).

Refer to Appendix F for sample cases of the evaluation of these integrals.

The element area integrals in equation (7.10) can be evaluated via Gauss-Legendre numerical

integration, which precludes the necessity for simplifications to the integrand. However, in the

discussions to follow, it will be useful to introduce acoustic far-field assumptions. See Kinsler

et al. [57] for a general description of these assumptions and Appendix D for details of this

particular implementation. These assumptions entail making the approximations

1

r1�

1

R

1

r01�

1

R(7.16)

and

ei!

�r1�r0

1

c

�� ei!

(�0��)K+(�0��)�c (7.17)

whereR is the distance from the center of the panel to the point(x; y; z) in the acoustic field,

K = sin � cos �, and� = sin � sin� (see Figure 7.1).

Equation (7.10) can be rewritten for the acoustic far-field as

p2t (R; �; �) =� �

2�R

�2 NeXi=1

NeXj=1

Z

Ai

Z

A0

j

fLw(�; �)gTi

1Z�1

!4[Sijaa(!)]ei!

(�0��)K+(�0��)�c d!

�Lw��0; �0

�jdAi dA

0

j

(7.18)

188

The out-of-plane interpolation functionsLw(�; �) are defined in terms of element local coordi-

nates in equation (4.4). Therefore, the complex exponential function in the integrand of equation

(7.18) must be separated into two components; one is constant for each element index pair(i; j),

the other must be included in the integration over the element areas in local coordinates. The

element area integrations can be performed numerically, via Gauss-Legendre numerical integra-

tion (see Appendix B), but significant improvements in computational efficiency can be realized

by performing the integrations in closed form (see Appendix C). Other acoustic quantities, such

as acoustic intensity and total radiated power, can be related to the mean square pressurep2t in

equation (7.18), shown in Appendix E.

7.2 Discrete Spectral Analysis

Recall the relationship between the radiated acoustic pressure at a point(x; y; z) in the

acoustic free-field and the dynamic response of the panel as described by Rayleigh’s integral

in equation (7.1). The physical degrees of freedom are known on an elemental basis from

equation (4.5)

we(�; �; t) � fLw(�; �)gTfabg (7.19)

wherefLw(�; �)g is a vector of out-of-plane interpolation functions andfabg is the vector of

out-of-plane degrees of freedom. So, the area integral in the expression for the radiated acoustic

pressure, equation (7.1), can be rewritten as the sum of element area integrals:

pt(x; y; z; t) =�

2�

NeX

i=1

Z

Ai

1

rfLw(�; �)g

T

i

n�ab

�t�

r

c

�oi

dAi (7.20)

whereNe is the number of elements andAi is the element area.

A discrete spectral analysis can be employed by writing the panel dynamic response and

radiated acoustic pressure as a sum of harmonic functions given by

nab

�t�

r

c

�o=

N!�1Xn=0

fabngei!n(t� r

c) (7.21)

189

and

pt(x; y; z; t) =N!�1X

n=0

pn(x; y; z)ei!nt (7.22)

whereN! is the number of discrete frequencies andfabng andpn are complex spectral amplitudes

of the out-of-plane nodal displacement response and radiated acoustic pressure, respectively.

Substitution of equations (7.21) and (7.22) in Rayleigh’s integral, equation (7.20), results in

N!�1X

n=0

pn(x; y; z)ei!nt = �

�

2�

NeX

k=1

Z

Ak

1

rfLw(�; �)g

Tk

N!�1Xn=0

!2nfabngk e

i!n(t� r

c)dAk (7.23)

This expression can be written for any one of the discrete frequencies as

pn(x; y; z) = ��

2�

NeXk=1

Z

Ak

1

rfLw(�; �)g

Tk !2

nfabngk e�i!n

r

c dAk (7.24)

If the excitation is deterministic, the expression for the radiated pressure complex spectral

amplitudes can be used in an inverse FFT, equation (7.22), to determine the radiated pressure

time response:

pt(x; y; z; t) = ��

2�

N!�1Xn=0

NeXk=1

Z

Ae

1

rfLw(�; �)g

Tk !2

nfabngk e�i!n

r

c dAk ei!nt (7.25)

The vectorfabngk is an element-k subset of the physical displacement spectral amplitude vector

given by equations (6.41) and (6.59):

fAbng = [�]dH(!n)cffng (7.26)

However, for stochastic excitation, a statistical approach must be used.

The cross-correlation function of the radiated acoustic pressure can be formulated from

equation (7.22) through the following expectation

Cpp�x; y; z; t1; x

0; y0; z0; t2�=

N!�1Xn=0

N!�1Xm=0

E[p�mpn]ei(!nt2�!mt1) (7.27)

190

The cross-correlation function can be expressed in terms of the panel dynamic response by

substituting from equation (7.24) as

Cpp

�~x; t1; ~x

0; t2�=� �

2�

�2 NeXk=1

NeXl=1

Z

Ak

Z

A0

l

1

r1r2fLw(�; �)g

Tk

N!�1Xm=0

N!�1Xn=0

!2m!2nE[fa�bmgkfabng

Tl ]

e�i!m(t1�r1

c)ei!n(t2�

r2

c)�Lw��0; �0�l dAk dA

0

l(7.28)

wherer1 is the distance from(�; �) on the panel to~x = (x; y; z) in the acoustic field andr2

is the distance from(�0; �0) on the panel to~x0 = (x0; y0; z0) in the acoustic field. The term

E[�a�bm

k

�a�bn

Tl] contains the elementk and l components of the generalized spectral matrix

of the physical coordinate displacement response.

Recall that the expected value of the spectral amplitudes takes a special form for stationary

response, so the generalized spectral matrix simplifies to

E[fa�bmgkfabngTl ] = E[fa�bmgkfabng

Tl ]�mn (7.29)

where �mn is the Kronecker delta. Then, the stationary radiated pressure cross-correlation

function resultsCpp(~x1; t1; ~x2; t2) ! Cpp(~x1; ~x2; � ), which can be written as

Cpp(~x1; ~x2; �) =N!�1Xn=0

E[p�mpn]ei!n(�+

r1�r2

c) (7.30)

where� = t2 � t1 is the time lag. This expression can be further simplified for zero time lag

(� = 0), which should not be confused with zero retarded time lag. Thus, the mean square

radiated acoustic pressure at a point (x; y; z) in the acoustic field becomes

p2t (x; y; z) =� �

2�

�2 NeXk=1

NeXl=1

Z

Ak

Z

A0

l

1

r1r01fLw(�; �)g

Tk

N!�1Xn=0

!4n[Gklaa(!n)]e

i!n

�r1�r

0

1

c

��Lw

��0; �0

�ldAkdA

0

l

(7.31)

wherer1 andr01 are distances from the same point (x; y; z) in the acoustic field to different points

on the panel,(r1 � r01)=c is the retarded time difference, and�Gklaa(!n)

�contains the element

191

k and l entries of the single-sided, stationary, physical displacement spectral matrix[GAA(!n)]

inferred from equation (6.59):

[GAA(!n)] = [�]dH�(!n)c�Gff(!n)

�dH(!n)c[�]

T (7.32)

Although the following discussion is analogous to that for acoustic radiation using the

continuous spectral approach, a brief description will be reiterated here for convenience and

because the resulting relations are slightly different for the discrete spectral analysis. The element

area integrals can be evaluated via Gauss-Legendre numerical integration, which precludes the

necessity for acoustic far-field approximations. However, in the discussions to follow, it will

be useful to introduce acoustic far-field assumptions for acoustic intensity calculations. See

Appendix D for a detailed discussion. These assumptions entail making the approximations

1

r1�

1

R

1

r01

�1

R(7.33)

and

ei!�r1�r0

1

c

�� ei!

(�0��)K+(�0��)�c (7.34)

where R is the distance from the panel center to the point(x; y; z) in the acoustic field,

K = sin � cos �, and� = sin � sin�, shown in Figure 7.1.

The mean square radiated acoustic pressure can be written for the acoustic far-field as

p2t (R; �; �) =� �

2�R

�2 NeXk=1

NeXl=1

Z

Ak

Z

A0

l

fLw(�; �)gTk

N!�1Xn=0

!4n[G

klaa(!n)]e

i!n(�0��)K+(�0��)�

c

�Lw

��0; �0

�ldAk dA

0

l

(7.35)

Note that the interpolation functionsLw(�; �) are usually known in terms of element local

coordinates. Therefore, the complex exponential function in the integrand of equation (7.35)

must be separated into two components: one is constant for each element index pair(k; l); the

other must be included in the integration over the element areas in local coordinates. The element

192

area integrations can be performed numerically, via Gauss-Legendre numerical integration (see

Appendix B), but significant improvements in computational efficiency can be realized by

performing the integrations in closed form (see Appendix C). Other acoustic quantities, such

as the acoustic intensity and total radiated power, can be related to the mean square pressure in

equation (7.35), shown in Appendix E.

In general, the expression in equation (7.35) must be used to calculate the far-field mean

square acoustic pressure radiated by a panel. However, recall the simplifications presented,

starting with equation (6.61), for the excitation class which includes acoustic pressure, single

or multiple equivalent concentrated forces, and base acceleration. A general expression for the

single-sided, physical displacement spectral matrix at discrete circular frequency!n is inferred

from equation (6.62):

[GAA(!n)] = fH�

A(!n)gfHA(!n)gT (7.36)

Substituting into the expression for the far-field mean square pressure and changing the order

of the summations results in

p2t (R;�; �) =� �

2�R

�2 N!�1Xn=0

NeXk=1

Z

Ak

fLw(�; �)gT

ke�i!n

�K+��

c dAk !2

nfH�

a(!n)gk

NeXl=1

!2nfHa(!n)g

T

l

Z

Al

fLw(�0; �0)gl e

i!n�0K+�0�

c dA0

l

=� �

2�R

�2 N!�1Xn=0

p�npn

(7.37)

where

pn =NeXl=1

!2nfHa(!n)gT

lf'(!n)g (7.38)

andf'(!n)g contains the integrated shape functions with the complex exponential. It is obvious

that the double element sum has been rendered a single element sum for excitations of this

type. Enormous computational savings result from this simplification. Additional savings can

be achieved by formingf'(!n)g from the closed form integrations in Appendix C.

193


In the previous sections, forms for the far-field mean square pressure radiated by a flexible

panel-type structure supported in an infinite baffle and driven by acoustic pressure or concentrated

force excitation were developed from Rayleigh’s integral using continuous and discrete spectral

analysis methods. The formulation will now be validated through comparison with fundamental

cases and experiment in the following two sections.

7.3.1 Simply-Supported Aluminum Panel

Comparisons will be made with the classical formulation developed by Roussos [52] for the

case of a simply-supported aluminum panel. Consider a simply-supported aluminum panel with

dimensions of 0.3556x0.254x0.001 m (14�10�0.040 inches). The material properties for the

specimen are shown in Table 7.1.

Table 7.1 Material properties for aluminum panel specimen.




Mass Density 2.767e+3 kg/m3 2.588e-4 lbf s2/in4

Coeff. of Thermal Exp. 22.3��/�C 12.4��/�F

As a first example, let the excitation be characterized by normally-incident, planar acoustic

waves. The acoustic excitation had a spectral level of 1.52 Pa2/Hz (3.2e-8 (psi)2/Hz) over a

bandwidth of 10–1000 Hz with a spectral resolution of 0.5 Hz. The panel was modeled by a

finite element mesh of 14x10 elements. Thirty modes were used in both the classical and finite

element models with a uniform modal critical damping ratio of 0.02.

A comparison of the tangential versus normal radiated acoustic intensity predicted by the

classical and finite element methods is shown in Figure 7.2. The plot is for a radial distance of

3.05 m (120 inches), polar angles of�=[0,�/2] with a �/18 increment, and an azimuthal angle of

194

�=0. The same plot, but only considering the single frequency component at 800 Hz, is shown

in Figure 7.3. It can be seen that the broadband frequency content tends to make the acoustic

radiation directivity more diffuse as expected. The classical and finite element analyses show

excellent agreement in both cases.

The transmission loss versus frequency for this case is shown in Figure 7.4. The usual

behavior of a dip at the fundamental frequency separating the stiffness-controlled region from

the resonance-controlled region can be easily seen. The upper end in frequency is approaching the

mass-controlled region of the transmission loss. The classical and finite element analyses show

excellent agreement. The transmission loss curves are indistinguishable up to approximately

800 Hz.

Now consider planar acoustic excitation with the following angles of incidence (�i=60�,

�i=45�). The excitation has a spectral level of 1.52 Pa2/Hz (3.2e-8 (psi)2/Hz) over a 10–1000

Hz bandwidth with a spectral resolution of 0.5 Hz. The plot in Figure 7.5 presents the tangential

versus normal component of the radiated acoustic intensity at a radial distance of 3.05 m (120

inches), polar angles of�=[0,�/2] with a �/18 increment, and an azimuthal angle of�=0. The

same plot for the single frequency component at 800 Hz is shown in Figure 7.6. These results

further demonstrate the good agreement between the classical and finite element analyses. The

transmission loss versus frequency for this case is shown in Figure 7.7. The antisymmetric

modal resonances pointed out in Figure 6.7 are seen as tiny dips in the transmission loss curve.

The acoustic radiation analysis using the FE method for structural response has been shown

to produce very accurate results in comparison with classical analysis. The analysis will now

be correlated with experimental measurement.


The experimental case considered in this section consists of a clamped aluminum panel

195

It⋅Sin(θ), lbf/(in⋅s)

I t⋅Co

s(θ)

,lbf

/(in

⋅s)

0 7E-070

7E-07

Classical 10-1000 HzFE 10-1000 Hz

Figure 7.2 Normal versus tangential radiated acoustic intensity R=3.05 m(120 inches),�=[0,2�], �=0 for normal acoustic incidence, 10–1000 Hz.

196

It⋅Sin(θ), lbf/(in⋅s)

I t⋅Co

s(θ)

,lbf

/(in

⋅s)

0 1.5E-100

1.5E-10

Classical 800 HzFE 800 Hz

Figure 7.3 Normal versus tangential radiated acoustic intensity R=3.05 m(120 inches),�=[0,2�], �=0 for normal acoustic incidence, 800 Hz.

197

Frequency, Hz

Tra

nsm

issi

on

Los

s,dB

0 250 500 750 10000

10

20

30

40

50

60


Figure 7.4 Transmission loss versus frequency for normal acoustic incidence, 10–1000 Hz.

198

I⋅sin(θ), lbf⋅in/s

I⋅co

s(θ)

,lbf

⋅in/s

0 1E-060

1E-06

Classical 10-1000 HzFE 10-1000 Hz

Figure 7.5 Normal versus tangential radiated acoustic intensity R=3.05 m (120inches),�=[0,2�], �=0 for �i=60, �i=45 acoustic incidence, 10–1000 Hz.

199

I⋅sin(θ), lbf⋅in/s

I⋅co

s(θ)

,lbf

⋅in/s

0 1E-100

1E-10

Classical 800 HzFE 800 Hz

Figure 7.6 Normal versus tangential radiated acoustic intensity R=3.05 m(120 inches),�=[0,2�], �=0 for �i=60, �i=45 acoustic incidence, 800 Hz.

200

Frequency, Hz

Tra

nsm

issi

on

Los

s,d

B

0 250 500 750 10000

10

20

30

40

50

60


Figure 7.7 Transmission loss versus frequency for�i=60,�i=45 acoustic incidence, 10–1000 Hz.

201

Table 7.2: Receiving microphone array measurementlocations, shown schematically in Figure 6.26.

Mic PositionRadial

Distance (m)Angle (deg) Mic Position

RadialDistance (m)

Angle (deg)

R1 0.6096 0 R5 0.6096 -30

R2 1.2192 0 R6 1.2192 -30

R3 0.6096 30 R7 0.9144 -45

R4 1.2192 30

subjected to normally incident, random acoustic excitation. Comparisons will be made between

predicted and measured radiated acoustic pressures in a transmission loss configuration.

7.4.1 Clamped Aluminum Panel

Experiments were conducted to measure the structural-acoustic response of an aluminum

panel subject to broadband, normal acoustic excitation. The experiments were performed in the

Transmission Loss Apparatus (TLA) at NASA Langley Research Center, shown schematically in

Figure 6.26. The experimental apparatus is described in some detail in the validation section of

Chapter 6 and in Turner and Rizzi [55]. Comparisons of the predicted and measured panel normal

velocity are shown in the previous chapter. Acoustic radiation results will be presented here.

The transmitted acoustic pressure was measured using a single vertical microphone array

consisting of three 1.27�10–2 m (half-inch) microphones; one at the height of the panel center,

one 0.3048 m (12 inches) above the center mic, and one 0.3048 m (12 inches) below the center

mic. Transmitted acoustic pressure autospectra were collected at 21 locations by positioning the

three-microphone array in each of seven measurement locations, defined in Table 7.2 and shown

schematically in Figure 6.26.

The power spectral density of the pressure incident upon the panel is shown in Figure 6.27.

Comparisons of the predicted and measured transmitted pressure are shown in Figures 7.8 and 7.9

for two measurement locations; radially perpendicular to the panel center and at an oblique angle.

The agreement is very good, particularly near the panel resonances. Theoretically, the pressure at

202

the top and bottom microphones of the array should be the same, due to symmetry. However, the

measured spectra showed an increase in the levels between the peaks with decreasing distance

to the hard floor in the receiving room. Thus, some of the discrepancies are attributable to

receiving room reflections. Also note that the non-symmetric modes, shown in the previous

panel response data in Chapter 6, do not radiate, as expected.

203

Frequency, Hz

Pre

ssur

eP

SD

,(p

si)2 /H

z

100 200 300 400 50010-15

10-14

10-13

10-12

10-11

10-10

10-9


Figure 7.8 Comparison of the predicted and measured transmitted pressure PSDat the center microphone of array location R1 (see Table 7.2 and Figure 6.26).

204

Frequency, Hz

Pre

ssur

eP

SD

,(p

si)2 /H

z

100 200 300 400 50010-15

10-14

10-13

10-12

10-11

10-10

10-9


Figure 7.9 Comparison of the predicted and measured transmitted pressure PSD atbottom microphone of array location R3 (see Table 7.2 and Figure 6.26).

205

Chapter 8Applications and Demonstrations

The tools developed and validated in the previous chapters will now be employed to

demonstrate the static and dynamic thermomechanical behavior of SMAHC structures and

compare performance gains achieved by embedded SMA actuators to conventional response

abatement treatments. Results will be presented for SMAHC beam and panel structures. Early

work performed in this study [16] showed that changes in SMA volume fraction have more

effect on structural control authority than prestrain level. More recent work has corroborated this

statement and shown that the SMA volume fraction is more easily varied/controlled in fabrication

than the prestrain level. Finally, it is known that the thermomechanical fatigue performance of

Nitinol actuators degrades substantially at prestrains greater than 5%. The results shown in this

chapter will focus on conservative prestrain levels of 4% for the beam specimens and 5% for the

panel specimens. For simplicity, a glass-epoxy matrix material system with constant material

properties will be used in all cases. Mechanical property data for Nitinol from Cross et al.

[28] will also be used. The material properties shown in Table 8.1 were used to generate the

results shown in this chapter. Note that these predictions, using material properties for Nitinol

from Cross et al. [28], are conservative relative to the properties measured from the Nitinol

material used previously in this study, i.e., stable recovery stress for Nitinol used in this study

at 4% prestrain is approximately 413 MPa (59.9 ksi) as compared to 310 MPa (45 ksi) for the

corresponding case in Cross et al. [28].

206

Table 8.1 Material properties for glass-epoxy and Nitinol.

Glass/Epoxy Nitinol

E1 53.78 GPa (7.8 Msi) As 32.22�C (90�F)

E2 17.93 GPa (2.6 Msi) Af 62.78�C (145�F)

G12 8.62 GPa (1.25 Msi) �r from Figure 2.3

�12 0.25 E from Figure 2.4

� 2031.8 kg/m3 � 0.3

(0.19�10-3 lb s2/in4)

�1 5.4�10-6 /�C (3.5�10-6 /�F) � 6450 kg/m3 (0.6067�10-3 lbf s2/in.4)

�2 30.6�10-6 /�C (11.4�10-6 /�F) �1a 6.61�10-6 /�C (3.67�10-6 /�F), T<As

from equation 2.4, T≥As

�2a 6.61�10-6 /�C (3.67�10-6 /�F), T<As

11.0�10-6 /�C (6.11�10-6 /�F), T>Af

interpolation, As≤T≤Af

Recall Figure 2.3 characterizing the recovery stress for Nitinol from Cross et al. [28]. The

reverse transformation (austenitic) characteristic temperatures, under free recovery conditions, for

the material represented in this plot areAs �35�C (95�F) andAf �50�C (120�F). Note from

Figure 2.3 thatAf is increased by mechanical constraint and increasing prestrain, whileAs is not

significantly affected by either. Furthermore, the “knee” in the recovery stress curve becomes less

distinct with increasing prestrain. These are well-known phenomena and are attributable to stress

increasingly inhibiting the formation of austenite and completion of the reverse transformation.

These insights will be useful in later discussions.

8.1 General Observations

The physical behavior of SMA actuators embedded in a composite matrix is somewhat

non-intuitive and is easily misinterpreted, so some general comments are deemed appropriate

here. References will be made to Figures 8.1a–8.1d, which show the non-deformed/reference

(gray-shade) and deformed (mesh) in-plane geometries of 0.457x0.025x0.001 m (18x1x0.040

207

in.) conventional and SMAHC beams with a lamination of (45/0/-45/90)s and cantilevered or

clamped-clamped boundary conditions. Consider the case of a cantilevered, conventional beam

(no SMA). As the beam is heated, with uniform temperature, it will expand everywhere except

at the boundary and will be stress-free at locations away from the boundary, shown in Figure

8.1a. The thermal force that is generated is thereby relieved by in-plane deformation. If the other

end is constrained, the thermal force in the axial direction cannot be relieved and a compressive

in-plane load is generated. This load will increase with temperature until a critical value is

reached, at which point the beam will buckle. Note that the thermal load in the transverse

direction is balanced by reaction forces at the boundary and relieved by in-plane deformation in

the transverse direction at locations away from the boundary (see Figure 8.1b). These physical

phenomena can also be interpreted from the expression for the resultant internal force given by

equation (2.38). As the cantilevered beam is heated, the thermal forceN�Tx is balanced by the

force due to in-plane deformationNmx such that the resultant forceNx is zero. Conversely, a

nonzero resultant force exists for the clamped-clamped case because of an imbalance between

N�Tx andNmx. The imbalance is maintained after buckling by contributions fromN�x.

An analogous, but inverse, situation exists for a SMAHC consisting of glass-epoxy with

uniformly distributed SMA in all layers (volume fraction of 30% and initial strain of 4%).

The thermal force in this case has a sense opposite of that expected because the effective

CTE is contractile rather than expansive. A cantilever beam of this type would contract

everywhere except at the boundary and will be stress free at locations away from the boundary,

exemplified in Figure 8.1c. The axial thermal force is again relieved by in-plane deformation (i.e.,

N�Tx = Nmx). Imposing a constraint at the other end prevents the axial in-plane deformation

and a substantial tensile in-plane force results. Note that the thermal forces (tensile) in the

transverse direction are again balanced by reactions at the boundary and transverse in-plane

deformation at locations away from the boundary, shown in Figure 8.1d. Development of the

208

x+1000*u, m0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

y+1

000

*v,m

a)

b)

c)

d)

Figure 8.1 Non-deformed/reference (gray-shade) and deformed (mesh) in-plane geometries for0.457x0.025x0.001 m (18x1x0.040 in.) cantilevered and clamped-clamped glass-epoxy andSMAHC beams. Displacements calculated at 65.56�C (150 �F) and magnified 1000 times.

209

tensile thermal force in the presence of mechanical constraints is the desirable attribute that is

being explored for adaptive stiffening of composite structures by embedded SMA actuators.

It is noted that in the previous discussions the arguments regarding SMAHC beams with

free boundaries are, strictly speaking, only qualitatively correct because the SMA material

properties used, Figure 2.3, result from constrained recovery measurements. However, the in-

plane displacements associated with the tensile recovery forces are small relative to the prestrain

imparted to the SMA actuators, so the predictions should be reasonably accurate for such cases.

Also note that a stiffening effect does result in this particular case of a SMA hybrid cantilever

because the modulus of the SMA is superior to that of the matrix material at temperatures where

the SMA is in the austenitic state. Additional insights will now be gleaned from a series of

examples.

8.2 Thermal Buckling and Post-buckling

Consider a glass-epoxy beam of 0.457x0.025x0.001 m (18x1x0.040 in.) dimensions with a

lamination of (45/0/-45/90)s and clamped boundaries at both ends. The critical temperature

change for this structure with a uniform temperature distribution is 1.68�C (3.03�F). The

normalized maximum post-buckling deflection versus temperature change from ambient for the

conventional beam (va=0%) is shown in Figure 8.2 in comparison with SMAHC beams. The

SMAHC beam responses correspond to the case of a 30% uniformly distributed volume fraction

of SMA with an initial strain of 4%. The symbolsva andAs indicate SMA volume fraction,

prestrain level, and austenite start temperature, respectively.

It can be seen that two buckling temperatures exist for the SMAHC beam withAs=32.22�C

(90�F); the first occurs at a temperature very near that for the conventional beam and the second at

a much higher temperature. This phenomenon is attributable to the fact thatAs, for this particular

alloy, is greater than the ambient temperature of 21.11�C (70�F) and can be explained as follows.

210

At temperatures less thanAs, the SMA is not activated and behaves like a conventional linear

thermoelastic material. So the SMAHC beam buckles at a critical temperature belowAs. The

SMA becomes activated atAs and the recovery stress rapidly increases with temperature, which

overcomes the compressive thermal stress and returns the beam to a flat configuration. At higher

temperatures, the rate of increase of the recovery stress with temperature diminishes and may

be surpassed by the rate of increase in matrix compressive thermal stress. Further increases

in temperature will cause net-compressive stress development, eventually leading to a second

incidence of thermal buckling.

In contrast, theAs=15.56�C (60�F) case in Figure 8.2 exhibits only a single thermal buckling

event at high temperature. This case was produced using the same SMA properties as before by

shifting the temperature scale associated with those properties –16.67�C (–30�F). This can easily

be achieved in practice by increasing the Nickel content of the alloy as little as 0.125 weight %

(see Buehler and Wang [58]). Note that the buckling temperature is slightly higher in this case

relative to theAs=32.22�C (90�F) case. This result is due to the fact that the shift in the SMA

property temperature scale for theAs=15.56�C (60�F) case produces a slightly higher recovery

force at the imposed temperature, which can be seen by referring to Figure 2.3.

Recall that the former effect, near-ambient buckling of a structure then complete arrestment of

the thermal post-buckling deflection by SMA recovery stress, was demonstrated experimentally

and correlated with prediction in Chapter 5 for the case of a clamped-clamped SMAHC beam.

Proof of this behavior has a numerical basis also, which will be explained subsequently. There

have been differing theories about the elimination of the thermal post-buckling deflection, which

results in two buckling temperatures forAs greater than ambient. The formulation by Turner,

Zhong, and Mei [16], based upon the RSM of the SMA constitutive behavior, corroborates the

results presented here. A different result was reported in Mei, Zhong, and Turner [59] where

an incremental approach with many temperature increments was deemed necessary to account

211

∆T, oC

Wm

ax/h

0 20 40 60 80 100 120 140 1600

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

va=0%va=30%, As=32.22 oCva=30%, As=15.56 oC

Figure 8.2 Normalized mid-span post-buckling deflection versus temperature change forconventional and SMAHC 0.457x0.025x0.001 m (18x1x0.040 in.) clamped beams.

212

for the SMA material nonlinearity in arriving at the final temperature. That formulation is also

based upon the RSM, but does not exhibit the complete elimination of thermal post-buckling

deflection for cases withAs greater than ambient. The difference between the two methods

was attributed to a mismatch of recovery stress and out-of-plane deflection terms in a thermal

post-buckling solution with multiple temperature increments based upon the formulation given

in Turner, Zhong, and Mei [16]. The ECTEM of the SMA constitutive behavior, presented

in this work, shows that the use of small temperature increments is not necessary because the

model inherently takes the material nonlinearity into account. Furthermore, any mismatch in

terms associated with multiple temperature increments in the thermal post-buckling solution can

be avoided by using a single temperature increment. The results of this approach are shown in

Figure 8.3 in comparison to the corresponding multi-step solution from Figure 8.2. The number

of temperature steps used in each post-buckling solution is indicated in the figure byN�T . Each

discrete symbol represents a separate solution achieved by initiating a post-buckling solution at

the first buckling temperature with a single temperature increment. These results show that the

present solution is insensitive to the number of temperature increments and complete elimination

of thermal post-buckling deflection for cases withAs greater than ambient and a flat, stress free

initial configuration is a general theoretically valid result. The ability to use a single temperature

increment can result in a huge computational savings also, particularly for high temperatures

relative to the buckling temperature.

Recall that the SMAHC beams considered thus far employed a uniformly distributed SMA

volume fraction of 30%. It is intuitive that performance gains could be achieved by strategically

placing the SMA actuators. One example of this technique is shown in Figure 8.3, where the

thermal post-buckling deflection is shown for a SMAHC beam with the same (45/0/-45/90)s

lamination but SMA actuators in the 0� layers only. A 50% volume fraction in the two 0� layers

results in an overall volume fraction of 12.5%. This overall volume fraction is less than half

213

∆T, oC

Wm

ax/h

0 20 40 60 80 100 120 140 1600

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

va=30%, N∆T=350va=30%, N∆T=1va=12.5%, 0o layers

Figure 8.3 Normalized mid-span post-buckling deflection versus temperature change forSMAHC 0.457x0.025x0.001 m (18x1x0.040 in.) clamped beams with SMA in all/0� layers.

214

that of the beam with uniformly distributed actuators, but the performance is slightly improved.

Note that this approach is more practical from a fabrication point of view as well, particularly

for automated fabrication where one could imagine replacing strategic spools of prepreg with

SMA ribbon in a continuous consolidation fabrication operation.

The effect of changing the SMA volume fraction within the 0� layers is shown in the thermal

post-buckling results in Figure 8.4. It can be seen that a volume fraction of 2.5% is insufficient

to eliminate thermal post-buckling, but its effect in reducing the deflection is clearly observed.

Thermal post-buckling can be eliminated for a range of temperatures with as little as 7.5% SMA

and the case of 12.5% SMA exhibits deflection elimination for a broad range of temperatures.

Similar behavior is exhibited by panel-type specimens as demonstrated in Figure 8.5. This

figure shows the normalized center deflection as a function of temperature change for a glass-

epoxy panel with a lamination of (45/0/-45/90/0/90/-45/0/45), a varying SMA volume fraction,

and clamped boundary conditions. As in the latter beam specimen case, the SMA in the panel

specimens is assumed to reside in the 90� and outer 0� layers only. Although it is not shown, it is

noted that slight performance gain is achieved under these conditions relative to a corresponding

laminate with SMA in all layers such that the overall SMA volume fraction is the same (see

Turner [60]). This material system, i.e., with SMA in only a few layers, is consistent with

the thought of automated fabrication made in the penultimate paragraph. Also note that the

middle 0�-layer is necessary to separate the two 90� layers containing a substantial SMA volume

fraction, thereby avoiding consolidation problems during cure. The results in Figure 8.5 show

that a volume fraction of 4% is insufficient to eliminate thermal post-buckling in this case, but

its effect in reducing the deflection is clearly observed. As little as 13% SMA will eliminate

thermal post-buckling over a small range of temperatures; higher concentrations of SMA (e.g.,

22%) can greatly extend the temperature range. The importance of these effects is more clear

when considering the dynamic response of these structures.

215

∆T, oC

Wm

ax/h

0 20 40 60 80 100 120 140 1600

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

va=0%va=2.5%, 0o layersva=7.5%, 0o layersva=12.5%, 0o layers

Figure 8.4 Normalized mid-span post-buckling deflection versustemperature change for composite 0.457x0.025x0.001 m (18x1x0.040

in.) clamped beams with various SMA volume fractions.

216

∆T, oC

Wm

ax/h

0 50 100 1500

1

2

3

4

5

6

7

va=0%va=4%va=13%va=22%

As-To

Figure 8.5 Post-buckling deflection of SMAHC laminates versus�T andva.

217

8.3 Dynamic Response

The dynamic response is assumed to be forced by a normally-incident, random acoustic

pressure with an overall sound pressure level (OASPL) of 120 dB (ref 20�Pa) and a bandwidth

of 10–400 Hz. A uniform modal critical damping ratio of 1% was used in the dynamic response

calculations. The maximum displacement response power spectral density (PSD) of the SMAHC

panel is shown in comparison to that of the baseline panel in Figure 8.6. It is clear that the

embedded SMA has little effect at ambient temperature. At 82�C (180�F,�T=61�C), however,

substantial benefit from the SMA recovery stress is evident in shifting the fundamental frequency,

shifting other modes out of bandwidth, and reducing peak responses. Thus, performance can

be enhanced by shifting resonance out of the excitation bandwidth and by response reduction

within the excitation bandwidth.

The thermal post-buckling observations have strong implications for the panels’ dynamic

response. An example is shown in Figure 8.7, where the maximum displacement PSD of

the SMAHC panel (va=22%) is shown at ambient and two elevated temperatures. At 82�C

(180�F, �T=61�C), substantial benefit of the embedded SMA is evident, while further heating

to 121�C (250�F, �T=100�C) greatly diminishes the performance (RMS Wmax=0.0927e-3 m)

because continued thermal expansion has partially overcome the beneficial effects of the recovery

stress. This particular material system, glass-epoxy matrix with 22% Nitinol volume fraction at

5% prestrain, achieves its “optimal” dynamic response at approximately 82�C (180�F), which

corresponds to the “knee” in the recovery stress curve (see Figure 2.3). This result also holds

for analogous SMAHC panels (va=22%) with embedded Nitinol having 3% and 4% prestrain,

which exhibit “optimal” dynamic performance at their respective “knee” temperatures of 54�C

(130�F) and 66�C (150�F). The optimal temperature for dynamic performance will be designated

by Ao and is governed by the same thermoelastic balance as that described in the thermal post-

buckling discussion. In general, with a different material system or at higher prestrain levels (in

218

Frequency, Hz

Dis

plac

emen

tPS

D,m

2 /Hz

0 100 200 300 40010-16

10-15

10-14

10-13

10-12

10-11

10-10

10-9

10-8

10-7

va=0%, ∆T=0oCva=22%, ∆T=0oCva=22%, ∆T=61oC

Figure 8.6Wmax PSD for the baseline and SMAHC laminates.

219

Frequency, Hz

Dis

plac

emen

tPS

D,m

2 /Hz

0 100 200 300 40010-16

10-15

10-14

10-13

10-12

10-11

10-10

10-9

10-8

10-7

va=22%, ∆T=0oCva=22%, ∆T=61oCva=22%, ∆T=100oC

Figure 8.7Wmax PSD for a SMAHC laminate at three temperatures.

220

the absence of a distinct “knee”),Ao must be determined numerically or by comparing the slope

of the recovery stress curve with that of the matrix thermal stress versus temperature.

Selection of an alloy composition and processing combination that will placeAo near the

application temperature is obviously recommended for dynamic control applications. An alternate

approach to achieve similar performance would entail increasing the SMA volume fraction and/or

prestrain, which will result in a less efficient and/or durable design. Requirements previously

cited for desirable static and dynamic performance imply potential for a delicate balance between

avoiding thermal buckling by imposingAs to be less than ambient temperature and “optimizing”

dynamic performance by specifyingAo to be near the application temperature. Thermal buckling

can also be avoided by boundary expansion, a common approach for aerospace structures, soAs

can be neglected in such cases andAo should be used to guide efficient design.

8.4 Sonic Fatigue and Acoustic Transmission

The performance of the SMAHC laminate will now be compared with conventional ap-

proaches including increased stiffness through additional composite layers and a constrained

layer damping (CLD) treatment. The conventional approaches were analyzed at ambient tem-

perature to simulate the conditions of boundary expansion, i.e., no thermal buckling. No attempt

was made to account for changes in matrix properties with temperature. A variety of dynamic

response performance characteristics are compiled in Table 8.2 for convenience of comparison.

The SMAHC panel achieves a 77% RMS displacement response reduction, relative to the base-

line 9–layer glass-epoxy panel, with a 48% mass increase. A conventional composite laminate of

13 layers (45/0/-45/90/45/0/90/0/45/90/-45/0/45) exhibits a similar mass increase (44%), but only

achieves a 61% RMS displacement reduction. A RMS displacement response of 0.0474e-3 m

(nearly equivalent to that of the SMAHC panel) can be achieved with a 16–layer (45/0/-45/90)2s

conventional laminate, but at the expense of a 78% mass increase. A typical CLD treatment

221

Table 8.2: Results from baseline, SMAHC, and panelswith conventional response abatement treatments.

CaseAddedMass

f1, HzRMS Wmax,

mTL, dB

Max RMS��

@ 120dBLife @150 dB

Baseline 0% 87.48 0.2062e-3 17.9 42.0 63.7e+3

SMAHC @ 82�C 48% 226.4 0.0476e-3 14.3 12.0 205e+6

13-Layer 44% 128.3 0.0805e-3 19.1 23.3 2.84e+6

16-Layer 78% 158.6 0.0474e-3 20.2 16.8 23.4e+6

Baseline-CLD 42% 73.4 0.0768e-3 27.4 15.6 37.8e+6

was modeled as a 19.16 N/m2 (0.2 lbf/ft2) weight addition and a uniform modal critical damp-

ing ratio of 6%, which are considered to be representative values. This treatment also happens

to result in a mass increase (42%) similar to that of the SMAHC, but the RMS displacement

response reduction of 63% is again inferior to the SMAHC. Additional insights are more clear

from comparisons of the displacement response spectra, as discussed subsequently.

A comparison of the respective maximum displacement response PSDs is shown in Figure

8.8. As anticipated, the means by which the three systems achieve dynamic response reduction

is quite different. The SMAHC achieves a much more weight-efficient stiffening effect relative

to that of adding layers, while the CLD treatment achieves reduction primarily by damping

enhancement. The weight-efficient stiffening effect of the embedded SMA is advantageous

for combination with other passive treatments, which are typically more effective at higher

frequencies. SMAHCs are also attractive for combination with control approaches involving

piezoelectric actuators, which typically exhibit actuation authority roll-off below 200 Hz. Such

combined active-passive approaches could achieve enormous dynamic response reductions by

using the embedded SMA to shift resonance frequencies and provide peak response control by

stiffness enhancement, while the active approach would provide enhanced damping at resonances

still within the excitation bandwidth.

The dynamic response results have direct implication for sonic fatigue life prediction.

222

Frequency, Hz

Dis

plac

emen

tPS

D,m

2 /Hz

0 100 200 300 40010-16

10-15

10-14

10-13

10-12

10-11

10-10

10-9

10-8

10-7

9-Layer, va=22%, 0.0476e-3 m RMS13-Layer, va=0%, 0.0805e-3 m RMS9-Layer, va=0%, CLD, 0.0768e-3 m RMS

Figure 8.8Wmax PSD comparison of two SMAHCs with conventional treatments.

223

Consider an analogous acoustic pressure loading condition with an OASPL of 150 dB (ref 20�Pa).

It is clear that linear dynamic response analysis is no longer valid, but its use at high response

levels is still common practice in the aerospace industry for sonic fatigue design. This condition

is more representative of a sonic fatigue environment and will produce more realistic life

predictions, while it is understood that conservative estimates will be obtained. A representative

strain-life (S-N) relationship for glass-epoxy laminates was modeled by� = 7385e�6(N)�0:1551

and the maximum RMS normal strain was taken as the failure criterion. In all cases, the

maximum strain was the transverse normal strain at the mid-span of the long boundary. Under

these conditions, the baseline panel has an expected life of 63.7k cycles. The corresponding

life predictions for the SMAHC, 13–layer conventional, and CLD panels are 205M, 2.84M,

and 37.8M cycles, respectively. These estimates are in direct relation with the corresponding

peak displacements in Figure 8.8. It is intuitive that measures to further attenuate the SMAHC

peak response would result in excellent sonic fatigue performance. The previously mentioned

active-passive system, for example, might be a particularly attractive solution.

A plot of the transmission loss (TL) versus frequency for the SMAHC panel at two

temperatures is shown in comparison to the baseline panel in Figure 8.9. The main effect

of the embedded SMA in its most effective state at 82�C (180�F) is a vast increase in the

stiffness-controlled portion of the TL. This improvement is diminished at higher temperatures

(e.g., 121�C) for the same reasons as stated above for the dynamic response. Although the

stiffness-controlled TL for the SMAHC panel shows dramatic improvement, the TL over the

400 Hz bandwidth is actually diminished because the mass-law effect has been moved to higher

frequencies. The TL for the SMAHC panel at 82�C (180�F) is shown in comparison to that for

the two conventional approaches in Figure 8.10. It can be seen that the SMAHC panel is superior

over a 200 Hz bandwidth, but not as effective at mid-range (resonance-controlled) frequencies

due to the shift in mass-law effects. Combination of the SMAHC with other passive and/or

224

active approaches is again an attractive alternative for this frequency range. At high frequencies

(mass controlled) the three approaches will produce nearly equivalent TL.

225

Frequency, Hz

Tra

nsm

issi

onLo

ss,d

B

0 100 200 300 4000

10

20

30

40

50

60

70

80

90

100

9-Layer, va=0%, ∆T=0oC, 17.9 dB9-Layer, va=22%, ∆T=61oC, 14.3 dB9-Layer, va=22%, ∆T=100oC, 16.1 dB

Figure 8.9 TL versus frequency for the baseline and SMAHC laminates.

226

Frequency, Hz

Tra

nsm

issi

onLo

ss,d

B

0 100 200 300 4000

10

20

30

40

50

60

70

80

90

100

9-Layer, va=22%, 14.3 dB13-Layer, va=0%, 19.1 dB9-Layer, va=0%, CLD, 27.4 dB

Figure 8.10 TL comparison of two SMAHCs with conventional treatments.

227

Chapter 9Summary and Concluding Remarks

The main objectives of this work were to develop a themomechanical model for predicting

the behavior of composite structures with embedded shape memory alloy (SMA) actuators and

to validate that model with experimental measurements. Other objectives were to develop SMA

actuator processing/characterization and SMA hybrid composite (SMAHC) fabrication techniques

for manufacturing structures with predictable and repeatable behavior. Finally, the experimental

results from the SMAHC specimens developed in this work were intended to demonstrate the

control authority of SMA actuators and demonstrate some of the complex behavior of SMAHC

structures.

A fairly extensive review of SMAs and their thermomechanical behavior was given in

Chapter 2 to introduce the topic and the demands on a constitutive model. A constitutive model

was then developed by a mechanics-of-materials approach to describe the behavior of a composite

lamina with an embedded SMA constituent. The thermoelastic model captures the material

nonlinearity of the SMA thermal strain/stress with temperature and is valid for constrained,

restrained, or free recovery behavior with appropriate measurements of fundamental engineering

properties. The constitutive model is also capable of modeling matrix material nonlinearity and

is in a form that is amenable to implementation in a commercial analysis environment. Classical

lamination theory was used to derive the resultant force and moment relations from the SMAHC

constitutive relations and von K´arman strain-displacement relations.

228

Fabrication and material system characterization processes for SMAHC specimens were

described in Chapter 3. The particular SMAHC material system used in this study consists of

a glass-epoxy, unidirectional, prepreg, tape with embedded Nitinol ribbon. Several practical

solutions for achieving repeatable results and overcoming practical fabrication issues were

demonstrated.

Extensive testing of the glass-epoxy and Nitinol materials was conducted to quantify the

behavior of a structure fabricated from the SMAHC material system. The mechanical properties

of the glass-epoxy in principal material directions (E1, E2 and G12, �1, �2) were determined

as a function of temperature from unidirectional and angle-ply tensile specimens. Isothermal

(modulus) and isolength (recovery force) tests were performed on Nitinol ribbon samples in

two conditions to determine the material properties as a function of temperature and thermal

cycle. Thermomechanical properties from the two conditions showed similar trends, but some

quantitative differences were evident. These differences were attributable mainly to aging effects

and stressed the importance of testing the actuator material in a condition that closely matches the

intended application as the actuator properties are very sensitive to condition and are a dominant

effect in the SMAHC performance.

In Chapter 4, the resultant force and moment relations were used along with the nodal degrees

of freedom and interpolation functions associated with a rectangular 24 degree-of-freedom plate

element to form a potential energy functional. The extended Hamilton’s principle was used to

derive the equations governing the response of a SMAHC plate-type structure to steady-state

thermal and dynamic mechanical loads. The governing equations were then separated into static

and dynamic components.

In Chapter 5, the equations governing static stability were derived within the framework

of the present formulation and used to develop a thermal buckling solution approach. The

iterative method was shown to produce valid solutions for an arbitrary material system with

229

nonlinear matrix material properties in addition to the nonlinearities associated with the SMA.

A thermal post-buckling solution method was also developed to predict the thermally deflected

shape subsequent to a thermal buckling event. An iterative Newton-Raphson approach was used

to account for geometric nonlinearities associated with large thermal deflections. The thermal

buckling and post-buckling solutions were validated against known solutions for conventional

structures to benchmark the formulation and implementation.

A SMAHC beam specimen was tested to determine its thermoelastic response in a clamped-

clamped configuration with a thermal load generated by resistive heating. The out-of-plane

deflection was measured as a function of temperature at the beam mid-span by a laser dis-

placement transducer. The predicted thermoelastic response was in excellent agreement with

the experimental measurement. Existing discrepancies were attributable to the effects of initial

deflection and mechanical boundary compliance in the physical system.

Continuous and discrete frequency domain methods were applied in Chapter 6 to the dynamic

equations of motion to predict geometrically linear dynamic responses of SMAHC plate-type

structures to various mechanical loads. The same approach was used in Chapter 7 with a

Rayleigh integral method to develop expressions for the radiated acoustic pressure resulting

from the dynamic motion of the structure. These solutions/implementations were validated

against classical solutions to benchmark the analyses. Further validation was performed with

experiment for conventional structures to assess the capability of the analyses to capture the

behavior of a physical system.

A SMAHC beam specimen was tested in a clamped-clamped configuration with a uniform

thermal load generated by resistive heating and band-limited random inertial excitation. Dynamic

response data was collected at ambient and at elevated temperatures from 32.22�C (90�F)

to 121.11�C (250�F) in 5.56�C (10�F) increments. The fundamental frequency of the beam

was shifted 430% at 121.11�C (250�F), relative to the ambient temperature resonance. The

230

maximum RMS displacement response attenuation of 84% was achieved at a temperature of

71.11�C (160�F) rather than at the maximum temperature. This was apparently caused by

enhanced damping due to stress induced martensitic transformation hysteresis. Nonetheless,

similar attenuations of around 80% were achieved at higher temperatures.

The predicted dynamic responses were in good agreement with experimental measurement

with the exception of a slightly over-predicted stiffening effect at elevated temperatures. This

discrepancy was again attributable to the fact that the mechanical boundaries were modeled as

perfectly rigid, when in fact the presence of thermal/electrical insulation in the gripping apparatus

resulted in significant boundary compliance. This effect was incorporated into the model, and

excellent agreement between prediction and measurement was achieved at all temperatures.

The model and analysis tools developed and validated in this work were then used in Chapter

8 to demonstrate a variety of thermomechanical response phenomena associated with SMAHC

structures and to compare the performance of a SMAHC panel-type specimen with conventional

dynamic response abatement techniques. Two buckling temperatures can exist for SMAHC

structures with a SMA constituent having an austenite start temperatureAs greater than ambient.

This effect results from the SMA recovery stress “pulling” the structure flat after initially buckling

at a temperature less thanAs and buckling again when continued thermal expansion overcomes

the recovery stress. The low-temperature buckling phenomenon can be avoided by selecting

a SMA with As less than ambient, thereby suppressing buckling altogether for a large range

of temperatures above ambient. This effect was later supported by experimental observations

described in Chapters 5 and 6.

Substantial improvements in performance can be realized by selectively placing the embed-

ded SMA in strategic locations such that the SMA volume fraction can be used more effectively,

which is also driven by fabrication considerations. Relatively modest changes in the volume

fraction in such cases can result in enormous changes in the performance. SMAHC structures

231

exhibit an “optimal” operating temperature (Ao) for dynamic response performance.Ao is a

function of the matrix material and SMA composition, volume fraction, and prestrain level. For

many casesAo corresponds to the “knee” temperature in the recovery stress curve. In general,

Ao must be determined numerically or by comparing the slope of the recovery stress curve with

that of the matrix thermal stress versus temperature. The implied conflict in choosing an alloy

for aerospace structures withAs < To to avoid thermal buckling andAo � T to “optimize”

dynamic response can be resolved by taking the conventional approach of allowing boundary

expansion to prevent buckling and usingAo � T to guide structural design.

The performance of a SMAHC panel was compared to that of a baseline panel and two

conventional approaches including additional conventional composite layers and a constrained

layer damping (CLD) treatment. The three response abatement approaches had similar added

mass relative to the baseline panel, but the SMAHC panel exhibited superior dynamic response

and sonic fatigue behavior. The SMAHC panel also exhibited superior transmission loss (TL)

characteristics in the low-frequency, stiffness-controlled region. The weight-efficient stiffening

effect of the embedded SMA makes it ideal for low-frequency, noise transmission abatement

and for combination with other passive or active approaches, which are typically more effective

at higher frequencies. These combined approaches have significant implications for dynamic

response/sonic fatigue also.

The major conclusions from the preceding discussions can be summarized as follows:

1) A practical constitutive model has been developed that accurately captures the mechanics of

a SMAHC material system.

2) Reliable SMAHC specimen fabrication procedures were developed and SMAHC beam

specimens were fabricated.

3) Experimental measurements from the SMAHC beam specimens demonstrated enormous

control over thermal buckling, post-buckling, and vibration responses.

232

4) Predicted static and dynamic responses using the model developed in this study were in

excellent agreement with the experimental measurements.

5) A SMAHC structure can exhibit two buckling temperatures. This phenomenon was supported

by experimental measurements.

6) SMAHC structures can exhibit an optimal operating temperature for dynamic response

performance.

7) SMAHC panel structures exhibit superior dynamic response characteristics to the conventional

approaches considered.

8) SMAHC structures are ideal for low-frequency TL and for combination with other treatments

for dynamic response and noise transmission abatement.

In conclusion, this study takes an engineering approach at considering embedded SMA

actuators as a means of achieving quasi–static control of composite structures through adaptive

stiffening in response to environmental (temperature) effects. The model was validated against

experimental measurement for static and dynamic responses. It is believed that implementation

of this model in a commercial FE code will lead to a very useful and practical analysis/design

tool. It is hoped that this study will provide useful information for the research and engineering

community in regard to fabrication, characterization, and modeling of SMAHC structures.

233

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240

Appendix ARSM Hybrid Lamina Constitutive Relations

The constitutive relations developed in this appendix are referred to as the recovery stress

model (RSM) as described in Chapter 2. A mechanics-of-materials approach is used to derive

the thermoelastic constitutive relations governing a SMAHC lamina, based upon observation of

the recovery stress phenomenon.

A representative volume element of a SMA fiber-reinforced hybrid composite lamina is

shown in Figure A.1. The element is taken to be in the plane of the plate. The composite

matrix, for example graphite-epoxy, has the principal material directions 1 and 2, and the SMA

fiber is embedded in the 1–direction.

σ2

σ1

Composite Matrix

Composite Matrix

SMA Fiber

2

1

Figure A.1 Representative volume element for SMA fiber-reinforced hybrid composite lamina.

In order to derive the constitutive relation for the 1–direction, it is assumed that a stress�1

acts alone on the element(�2 = 0) and that the SMA fiber and composite matrix are strained

by the same amount,�1 (i.e., plane sections remain plane). The 1–direction stress-strain relation

241

of the SMA fiber can be described as

�1a = Ea�1 + �r ; T � As (A.1)

or

�1a = Ea(�1 � �a4T ) ; T < As (A.2)

whereAs is the austenite start temperature and�a is the thermal expansion coefficient. The

Young’s modulusEa and the recovery stress�r are temperature dependent. The recovery stress�r

is also dependent on the initial strain�r. Description of the recovery stress and modulus behavior

for Nitinol is discussed in Chapters 2 and 3. Similarly, the one–dimensional stress–strain relation

in the 1–direction for the composite matrix can be expressed as

�1m = E1m(�1 � �1m4T ) (A.3)

The resultant force in the 1-direction(�2 = 0) is distributed over the SMA fiber and

composite matrix and can be written as

�1A1 = �1aAa + �1mAm (A.4)

where (�1; A1), (�1a; Aa), and (�1m; A1m) are the (stress, cross-sectional area) of the entire

element, SMA fiber, and composite matrix, respectively. Thus, the average stress�1 is

�1 = �1ava + �1mvm (A.5)

whereva = Aa=A1 and vm = Am=A1 are the volume fractions of SMA and composite matrix,

respectively. WhenT > As, the SMA effect is activated and the one-dimensional stress-strain

relation in the 1-direction becomes

�1 = (Ea�1 + �r)va + E1m(�1 � �1m4T )vm

= E1�1 + �rva � E1m�1mvm4T(A.6)

242

where

E1 = E1mvm + Eava (A.7)

WhenT < As, the SMA effect is not activated and the stress�1 is�1 = E1�1 � (Ea�ava + E1m�1mvm)4T

= E1(�1 � �14T )(A.8)

where

�1 =(E1m�1mvm + Ea�ava)

(E1mvm + Eava)(A.9)

A similar constitutive relation may be derived for the 2–direction by assuming that the

applied stress�2 acts upon both the fiber and the matrix(�1 = 0). Thus, the one-dimensional

stress-strain relations in the 2–direction for the SMA fiber and the composite matrix become

�2s = �2 = Ea(�2a � �a4T ) (A.10)

and

�2m = �2 = E2m(�2m � �2m4T ) (A.11)

respectively. The recovery stress does not appear in equation (A.10), since the SMA fiber initial

strain �r and recovery stress�r are considered to be a 1–direction effect only.

The total elongation is due to strain in the composite matrix and the SMA fiber and may

be written in the form

A1�2 = Am�2m +Aa�2a (A.12)

Thus, the total strain becomes

�2 = �2mvm + �2ava (A.13)

Since �2 = E2(�2 � �2�T ), equations (A.10) and (A.11) may be substituted into equation

(A.13) to give

�2 =�2

E2

+ �2�T

=�2va

Ea

+�2vm

E2m

+ (�ava + �2mvm)�T(A.14)

243

Therefore, the modulus and thermal expansion coefficient in the 2–direction become

E2 =

E2mEa

(E2mva + Eavm)(A.15)

and

�2 = �2mvm + �ava (A.16)

Expressions for the hybrid composite Poisson’s ratios and shear moduli follow from similar

derivations.

The constitutive relations for a thin composite lamina with embedded SMA fibers can be

derived using a similar engineering approach to give8<:�1�2�12

9=; =

24Q�

11Q�

120

Q�

12Q�

220

0 0 Q�

66

358<:�1�2 12

9=; +

8<:�r

00

9=;va � [Q]

m

8<:�1

�2

0

9=;m

vm4T

= [Q�]

8<:�1�2 12

9=;+

8<:�r00

9=;va �

0@[Q]

8<:�1

�2

0

9=;v1Am

4T ; T � As

(A.17)

and 8<:�1�2�12

9=; = [Q�]

0@8<:�1�2 12

9=;�

8<:�1

�2

0

9=;4T

1A ; T < As (A.18)

where [Q]m

and [Q�] are the reduced stiffness matrices of the composite matrix and the SMA

hybrid composite lamina, respectively. The[Q�] matrix is temperature dependent and is evaluated

using the previously derived relations as

(E1; �12) = (E1m; �12m)vm + (Ea; �a)va (A.19)

and

(E2; G12) =(E2mEa; G12mGa)

[(E2m; G12m)va + (Ea; Ga)vm](A.20)

where the�’s are Poisson’s ratios and theG’s are the shear moduli. The thermal expansion

coefficients�1 and �2 are derived in equations (A.9) and (A.16).

244

Appendix BGauss-Legendre Numerical Integration

B.1 Element Matrices

The element area integrals necessary for evaluation of the element matrices in equations

(4.56) can be performed numerically using Gauss-Legendre quadrature. The four-node, rectan-

gular, 24–dof plate element used in this study has interpolation functions given by equations

(4.3) and (4.4) for the in-plane and out-of-plane degrees of freedom, respectively. Consider the

element bending mass matrix for example, i.e.,

[mb] =

Z

Ae

�hfHw(x; y)gfHw(x; y)gTdAe

=

bZ

0

aZ

0

�hfHw(x; y)gfHw(x; y)gTdxdy

(B.1)

where (x; y) are the element coordinates anda and b are the element length and width,

respectively.

The element area integrations in equation (B.1) are in a form that is easily adapted to

Gauss-Legendre numerical integration. The coordinates can be transformed to non-dimensional

element-centered coordinates(�; �) with the variable transformation

x : [0; a] � = 2x

a� 1 : [�1; 1] d� =

a

2dx

y : [0; b] � = 2y

b� 1 : [�1; 1] d� =

b

2dy

(B.2)

Thus, the expression in equation (B.1) can be rewritten as

[mb] =

1Z

�1

1Z

�1

�hn

~Hw(�; �)on

~Hw(�; �)oT

jJ(�; �)j d�d� (B.3)

where jJ(�; �)j is the determinant of the Jacobian matrix of the transformation:

jJ(�; �)j =

��@x@�

@x@�

@y@�

@y@�

�� =��2

a0

0 2

b

�� = 4

ab(B.4)

245

1

34

(-1,-1) (1,-1)

(1,1)(-1,1)

ξ,u

η,v

ζ,w

2

Figure B.1 Schematic of plate element in centroidal non-dimensional coordinates.

The element displacement functions are now written in terms of the non-dimensional coordinates

(�; �) as

w(x; y) =4X

i=1

�Gi(�)Gi(�)wi +

a

2Hi(�)Gi(�)

�@w@x

�i

+b

2Gi(�)Hi(�)

�@w@y

�i

+ab

4Hi(�)Hi(�)

� @2w

@x@y

�4

� (B.5)

whereGi(�) =

1

4

��i�

3 + 3�i� + 2�

Hi(�) =1

4

��3 + �i�

2� � � �i

� (B.6)

and (�i; �i) are the coordinates of nodei. A schematic of an element in the non-dimensional

coordinates is shown in Figure B.1

The integral over the element are in equation (B.3) is now in a form that is easily evaluated

numerically. Gauss-Legendre integration entails evaluating the integrand at a number of equally

246

spaced (Gauss) points within the region of integration and summing the product of these

contributions with corresponding weights. Exact evaluation is possible for polynomial-type

functions when the order of the function is less than or equal to one less than twice the number

of Gauss points, i.e.,(2Ng � 1 � p) whereNg is the number of Gauss points andp is the order

of the polynomial. Employing this method, the integral in equation (B.3) can be rewritten as

[mb] =

NgX

i=1

NgX

j=1

WiWj�hn

~Hw(�i; �j)on

~Hw(�i; �j)oT

jJ(�i; �j)j (B.7)

whereW� are the weights at the Gauss points denoted by�i and �j.

B.2 Mean Square Radiated Acoustic Pressure

A similar approach can be used to evaluate the element area integrals for acoustic radiation

calculations in Chapter 7. Recall the form for the mean square radiated pressure from the

continuous spectral analysis approach in equation (7.18). In this case, the element physical

coordinates are denoted by(�; �) and the non-dimensional element-centered coordinates will be

denoted by(�; �). Equation (7.18) can be written in terms of the element physical coordinates

as follows

p2(R; �; �) =� �

2�R

�2 NeXi=1

NeXj=1

bZ

0

aZ

0

bZ

0

aZ

0

fHw(�; �)gTi

1Z�1

!4[Sijaa(!)]ei!

(�0��)K+(�0��)�c d!

�Hw

�� 0; �0

�jd� d� d�0 d�0

(B.8)

Applying the variable transformation given in equation (B.2) yields

p2(R; �; �) =� �

2�R

�2 NeXi=1

NeXj=1

1Z

�1

1Z

�1

1Z

�1

1Z

�1

n~Hw(�; �)

oT

i

1Z�1

!4[Sijaa(!)]ei!

(�0��)K+(�0��)�c d!

n~Hw

��0; �0

�ojjJ(�; �)j

��J��0; �0�� d� d� d�0 d� 0

(B.9)

where thejJ j are determinants of the Jacobian of the transformation as before. Now, applying

Gauss-Legendre numerical integration gives the final expression for the mean square pressure at

247

a point (i.e., for a given set of observer direction angles� and�) in the acoustic far-field:

p2(R; �; �) =� �

2�R

�2 NeXi=1

NeXj=1

NgXk=1

NgXl=1

NgXm=1

NgXn=1

WkWlWmWn

n~Hw(�k; �l)

oTi

1Z

�1

!4[Sijaa(!)]e

i!(�m��k)K+(�n��l)�

c d!n

~Hw(�m; �n)ojjJ(�k; �l)jjJ(�m; �n)j

(B.10)

The same procedure can be applied to the expression for the mean square radiated pressure from

the discrete spectral approach, equation (7.35), which yields

p2t (R; �; �) =� �

2�R

�2 NeXi=1

NeXj=1

NgXk=1

NgXl=1

NgXm=1

NgXn=1

WkWlWmWn

n~Hw(�k;�l)

oTi

N!�1Xr=0

!4rE[fa�brgifabrg

Tj ]e

i!r(�m��k)K+(�n��l)�

c

n~Hw(�m;�n)

ojjJ(�k;�l)jjJ(�m;�n)j

(B.11)

It is apparent that numerical integration for the computation of the radiated pressure can be

computationally intensive due to the multiply-nested loops. Clever computer coding and/or

compilation optimizers can diminish this problem, but another approach would be to evaluate

the necessary element area integrals in closed form. This can save significant computational

effort and a discussion of this approach is given in Appendix C.

B.3 Radiated Acoustic Power

Determination of the radiated acoustic power requires integration of the normal acoustic

intensity over a hemispherical surface, given by equation (E.6). This integral must be evaluated

numerically. Gauss-Legendre numerical integration can be employed through selection of a

suitable change of variables. The following change of variables results in the desired integral

limits:

� : [0; �=2] u =4

�

��

�

4

�: [�1; 1] d� =

�

4du

� : [0; 2�] v =1

�(�� ) : [�1; 1] d� = � dv

(B.12)

248

Then the radiated acoustic power becomes

� =

1Z

�1

1Z

�1

I(r; u; v) r2 sin��4u+

�

4

�jJ(u; v)j du dv (B.13)

where the determinant of the Jacobian matrix is as follows

jJ(u; v)j =

��@�@u

@�@v

@�@u

@�@v

�� =��4

00 �

�� = �2

4(B.14)

as is easily seen in making the substitution ford� andd� in equation (B.13) from the transfor-

mation. Applying Gauss-Legendre integration results in the following expression for the total

acoustic power radiated by the panel

� =

NgXr=1

NgXs=1

I(r; ur; vs) r2 sin

��4ur +

�

4

�jJ(ur; vs)j (B.15)

where the acoustic intensityI is related to the mean square radiated acoustic pressure by equation

(E.4) and the mean square pressure is given in equation (B.10) or (B.11).

249

Appendix CClosed Form Evaluation of Element Area Integrals

Recall the expressions for the far-field mean square radiated acoustic pressure, given in

equations (7.18) or (7.35):

p2t (R; �; �) =� �

2�R

�2 NeXi=1

NeXj=1

Z

Ai

Z

A0

j

fHw(�; �)gTi

1Z

�1

!4[Sijaa(!)]ei!

(�0��)K+(�0��)�c d!

�Hw

��0; �0

�jdAi dA

0

j

(C.1)

or

p2t (R; �; �) =� �

2�R

�2 NeXi=1

NeXj=1

Z

Ai

Z

A0

j

fHw(�; �)gTi

N!�1Xn=0

!4nE[fa�bngifabng

Tj ]e

i!n(�0��)K+(�0��)�

c

�Hw

��0; �0

�jdAi dA

0

j

(C.2)

The form for the out-of-plane interpolation functionsfHw(�; �)g for the 24 degree-of-freedom

rectangular plate element are given in equation (4.4). As stated previously in the respective

acoustic radiation sections, the complex exponential function in the integrand of the above

equations must be separated into two components. One is constant for each element index pair

(i; j) and the other must be included in the integration over the element areas in local coordinates.

Consider the expression for the mean-square pressure in the discrete frequency domain

formulation for example. Splitting the complex exponential into element-constant and element-

variable components and changing the order of the summations gives

p2t (R; �; �) =� �

2�R

�2 N!�1Xn=0

!4n

NeXi=1

NeXj=1

ei!n(�0j��i)K+(�0j��i)�

c

Z

Ai

fHw(u; v)gTi e

�i!nuK+v�

c dAi

E[fa�bngifabngTj ]

Z

A0

j

�Hw

�u0; v0

�jei!n

u0K+v0�

c dA0

j

(C.3)

250

where (u; v) are element local coordinates. There are four cases which must be considered

in evaluation of these area integrals;(K 6= 0;� 6= 0). (K 6= 0;� = 0), (K = 0;� 6= 0) and

(K = 0;� = 0). Integration of each of the 16 out-of-plane interpolation functions with the

complex exponential, for each of the four cases involving the argument of the exponential

function, was performed. The resulting functions are denoted as follows

I1(!n;K;�) =

aZ

0

bZ

0

M1(u; v)ei!n

uK+v�

c dudv

...

I16(!n;K;�) =

aZ

0

bZ

0

M16(u; v)ei!n

uK+v�

c dudv

I1(!n;K) =

aZ

0

bZ

0

M1(u; v)ei!n

uK

c dudv

...

I16(!n;K) =

aZ

0

bZ

0

M16(u; v)ei!n

uK

c dudv

I1(!n;�) =

aZ

0

bZ

0

M1(u; v)ei!n

v�

c dudv

...

I16(!n;�) =

aZ

0

bZ

0

M16(u; v)ei!n

v�

c dudv

I1 =

aZ

0

bZ

0

M1(u; v) dudv

...

I16 =

aZ

0

bZ

0

M16(u; v) dudv

(C.4)

wherea and b are the element length and width, respectively. The expressions resulting from

integration of the shape functions along with the various exponentials follow. Note that wave

251

number is used in these expressionsk = !=c.

(K 6= 0;� 6= 0)

I1(k;K;�) = �[��12i+ 12ieibk� + 6bk� + 6bk�eibk� + b3k3�3

�

��12i+ 12ieiakK + 6akK+ 6akKeiakK + a3k3K3

�]=�a3b3K4�4k8

� (C.5)

I2(k;K;�) = [��12i+ 12ieibk� + 6bk� + 6bk�eibk� + b3k3�3

�

��12i+ 12ieiakK + 6akK+ 6akKeiakK + a3k3K3eiakK

�]=�a3b3K4�4k8

�(C.6)

I3(k;K;�) = �[��12i+ 12ieibk� + 6bk� + 6bk�eibk� + b3k3�3eibk�

�

��12i+ 12ieiakK + 6akK+ 6akKeiakK + a3k3K3eiakK

�]=�a3b3K4�4k8

�(C.7)

I4(k;K;�) = [��12i + 12ieibk� + 6bk� + 6bk�eibk� + b3k3�3eibk�

�

��12i + 12ieiakK + 6akK+ 6akKeiakK + a3k3K3

�]=�a3b3K4�4k8

� (C.8)

I5(k;K;�) = �[��12i+ 12ieibk� + 6bk� + 6bk�eibk� + b3k3�3

�

��6i+ 6ieiakK + 4akK+ 2akKeiakK + ia2k2K2

�]=�a2b3K4�4k8

� (C.9)

I6(k;K;�) = [�12 � 12eibk� + 6ibk� + 6ibk�eibk� + ib3k3�3

�

�6� 6eiakK + 2iakK+ 4iakKeiakK + a2k2K2eiakK

�]=�a2b3K4�4k8

� (C.10)

I7(k;K;�) = [��12 + 12eibk� � 6ibk�� 6ibk�eibk� � ib3k3�3eibk�

�

�6� 6eiakK + 2iakK+ 4iakKeiakK + a2k2K2eiakK

�]=�a2b3K4�4k8

� (C.11)

I8(k;K;�) = [��12i+ 12ieibk� + 6bk� + 6bk�eibk� + b3k3�3eibk�

�


�]=�a2b3K4�4k8

� (C.12)

I9(k;K;�) = �[��6i+ 6ieibk� + 4bk� + 2bk�eibk� + ib2k2�2

�

��12i + 12ieiakK + 6akK+ 6akKeiakK + a3k3K3

�]=�a3b2K4�4k8

� (C.13)

I10(k;K;�) = [��6i+ 6ieibk� + 4bk� + 2bk�eibk� + ib2k2�2

�

��12i + 12ieiakK + 6akK+ 6akKeiakK + a3k3K3eiakK

�]=�a3b2K4�4k8

�(C.14)

252

I11(k;K;�) = [�6� 6eibk� + 2ibk�+ 4ibk�eibk� + b2k2�2eibk�

�

��12 + 12eiakK � 6iakK� 6iakKeiakK � ia3k3K3eiakK

�]=�a3b2K4�4k8

�(C.15)

I12(k;K;�) = [�6 � 6eibk� + 2ibk� + 4ibk�eibk� + b2k2�2eibk�

�

�12 � 12eiakK + 6iakK+ 6iakKeiakK + ia3k3K3

�]=�a3b2K4�4k8

� (C.16)

I13(k;K;�) = �[��6i+ 6ieibk� + 4bk� + 2bk�eibk� + ib2k2�2

�


�]=�a2b2K4�4k8

� (C.17)

I14(k;K;�) = [�6 � 6eibk� + 4ibk� + 2ibk�eibk� � b2k2�2

�

�6 � 6eiakK + 2iakK+ 4iakKeiakK + a2k2K2eiakK

�]=�a2b2K4�4k8

� (C.18)

I15(k;K;�) = [�6 � 6eibk� + 2ibk� + 4ibk�eibk� + b2k2�2eibk�

�

�6 � 6eiakK + 2iakK+ 4iakKeiakK + a2k2K2eiakK

�]=�a2b2K4�4k8

� (C.19)

I16(k;K;�) = [�6� 6eibk� + 2ibk� + 4ibk�eibk� + b2k2�2eibk�

�

�6� 6eiakK + 4iakK+ 2iakKeiakK � a2k2K2

�]=�a2b2K4�4k8

� (C.20)

(K 6= 0;� = 0)

I1(k;K) = b�12 � 12eiakK + 6iakK+ 6iakKeiakK + ia3k3K3

�=�2a3k4K4

�(C.21)

I2(k;K) = b��12 + 12eiakK � 6iakK� 6iakKeiakK � ia3k3K3eiakK

�=�2a3k4K4

�(C.22)

I3(k;K) = b��12 + 12eiakK � 6iakK� 6iakKeiakK � ia3k3K3eiakK

�=�2a3k4K4

�(C.23)

I4(k;K) = b�12 � 12eiakK + 6iakK+ 6iakKeiakK + ia3k3K3

�=�2a3k4K4

�(C.24)

I5(k;K) = b�6� 6eiakK + 4iakK+ 2iakKeiakK � a2k2K2

�=�2a2k4K4

�(C.25)

I6(k;K) = b�6� 6eiakK + 2iakK+ 4iakKeiakK + a2k2K2eiakK

�=�2a2k4K4

�(C.26)

I7(k;K) = b�6� 6eiakK + 2iakK+ 4iakKeiakK + a2k2K2eiakK

�=�2a2k4K4

�(C.27)

I8(k;K) = b�6� 6eiakK + 4iakK+ 2iakKeiakK � a2k2K2

�=�2a2k4K4

�(C.28)

253

I9(k;K) = b2�12 � 12eiakK + 6iakK+ 6iakKeiakK + ia3k3K3

�=�12a3k4K4

�(C.29)

I10(k;K) = b2��12 + 12eiakK � 6iakK� 6iakKeiakK � ia3k3K3eiakK

�=�12a3k4K4

�(C.30)

I11(k;K) = b2�12 � 12eiakK + 6iakK+ 6iakKeiakK + ia3k3K3eiakK

�=�12a3k4K4

�(C.31)

I12(k;K) = b2��12 + 12eiakK � 6iakK� 6iakKeiakK � ia3k3K3

�=�12a3k4K4

�(C.32)

I13(k;K) = b2�6� 6eiakK + 4iakK+ 2iakKeiakK � a2k2K2

�=�12a2k4K4

�(C.33)

I14(k;K) = b2�6� 6eiakK + 2iakK+ 4iakKeiakK + a2k2K2eiakK

�=�12a2k4K4

�(C.34)

I15(k;K) = b2��6 + 6eiakK � 2iakK� 4iakKeiakK � a2k2K2eiakK

�=�12a2k4K4

�(C.35)

I16(k;K) = b2��6 + 6eiakK � 4iakK� 2iakKeiakK + a2k2K2

�=�12a2k4K4

�(C.36)

(K = 0;� 6= 0)

I1(k;�) = a�12 � 12eibk� + 6ibk�+ 6ibk�eibk� + ib3k3�3

�=�2b3k4�4

�(C.37)

I2(k;�) = a�12 � 12eibk� + 6ibk�+ 6ibk�eibk� + ib3k3�3

�=�2b3k4�4

�(C.38)

I3(k;�) = a��12 + 12eibk� � 6ibk�� 6ibk�eibk� � ib3k3�3eibk�

�=�2b3k4�4

�(C.39)

I4(k;�) = a��12 + 12eibk� � 6ibk�� 6ibk�eibk� � ib3k3�3eibk�

�=�2b3k4�4

�(C.40)

I5(k;�) = a2�12 � 12eibk� + 6ibk� + 6ibk�eibk� + ib3k3�3

�=�12b3k4�4

�(C.41)

I6(k;�) = a2��12 + 12eibk� � 6ibk�� 6ibk�eibk� � ib3k3�3

�=�12b3k4�4

�(C.42)

I7(k;�) = a2�12 � 12eibk� + 6ibk� + 6ibk�eibk� + ib3k3�3eibk�

�=�12b3k4�4

�(C.43)

I8(k;�) = a2��12 + 12eibk� � 6ibk�� 6ibk�eibk� � ib3k3�3eibk�

�=�12b3k4�4

�(C.44)

I9(k;�) = a�6 � 6eibk� + 4ibk� + 2ibk�eibk� � b2k2�2

�=�2b2k4�4

�(C.45)

I10(k;�) = a�6� 6eibk� + 4ibk�+ 2ibk�eibk� � b2k2�2

�=�2b2k4�4

�(C.46)

I11(k;�) = a�6 � 6eibk� + 2ibk� + 4ibk�eibk� + b2k2�2eibk�

�=�2b2k4�4

�(C.47)

254

I12(k;�) = a�6 � 6eibk� + 2ibk� + 4ibk�eibk� + b2k2�2eibk�

�=�2b2k4�4

�(C.48)

I13(k;�) = a2�6 � 6eibk� + 4ibk� + 2ibk�eibk� � b2k2�2

�=�12b2k4�4

�(C.49)

I14(k;�) = a2��6 + 6eibk� � 4ibk�� 2ibk�eibk� + b2k2�2

�=�12b2k4�4

�(C.50)

I15(k;�) = a2��6 + 6eibk� � 2ibk�� 4ibk�eibk� � b2k2�2eibk�

�=�12b2k4�4

�(C.51)

I16(k;�) = a2�6� 6eibk� + 2ibk� + 4ibk�eibk� + b2k2�2eibk�

�=�12b2k4�4

�(C.52)

(K = 0;� = 0)

I1! I4 =ab

4(C.53)

I5; I8 =a2b

24I6; I7 = �

a2b

24(C.54)

I9; I10 =ab2

24I11; I12 = �

ab2

24(C.55)

I13; I15 =a2b2

144I14; I16 = �

a2b2

144(C.56)

These expressions were used as function calls to evaluate the vectors pre- and post-multiplying

the element-ij displacement spectral matrix in equation (C.3).

255

Appendix DRadiation Far-Field Approximations

Prediction of the radiated acoustic pressure to a free field from a vibrating surface requires

evaluation of Rayleigh’s integral, equation (7.1). The present work casts the radiated acoustic

pressure response in three forms: continuous spectral analysis of stochastic response, given by

equation (7.10); and discrete spectral analysis of deterministic and stochastic responses, given

by equations (7.24) and (7.31). The existence of the radial distance dependence in the integrand

requires the use of numerical integration or some simplifying assumptions. The integrand can be

simplified extensively by considering only far-field observer locations in the acoustic free-field.

Let the observation point(x; y; z) in the acoustic field be designated by its distanceR from the

panel center and two direction angles: the polar angle� = [0; �=2], and the azimuthal angle

� = [0; 2�]. Thus,

R =p

x2 + y2 + z2

x = R sin � cos �

y = R sin � sin�

z = R cos �

(D.1)

and the distance from a point on the panel to the observation point

r =

q(x� �)2 + (y � �)2 + z2 (D.2)

becomes

r = R

s1 �

2 sin � cos�

R� +

��

R

�2

�

2 sin � sin�

R� +

� �R

�2(D.3)

The acoustic far-field approximation entails neglecting the terms in the expression forr that

involve 1=R2 and letting

1

r�

1

R(D.4)

256

The remaining terms in the expression forr may be expanded in a binomial series:

r = R

�1 �

1

2

�2� sin � cos �

R+2� sin � sin �

R

�+O

�1

R2

�+ � � �

�(D.5)

Retaining the first two terms in the expansion results in

r � R

�1�

sin � cos �

R� �

sin � sin�

R�

�(D.6)

or

r � R�K� �� (D.7)

whereK = sin � cos� and � = sin � sin�. Thus, the exponential term in the integrand of

equation (7.24) can be simplified as follows.

e�i!nr

c � e�i!n

c(R�K��) (D.8)

Finally, the difference in radial distances encountered in equations (7.10) and (7.31) becomes

r1 � r01 � K��0 � �

�+ �

��0 � �

�(D.9)

and the corresponding exponential term in the integrand takes the following form.

ei!n

�r1�r

0

1

c

�� ei

!n

c[(�0��)K+(�0��)�] (D.10)

257

Appendix EAcoustic Intensity, Power, and TL

The acoustic intensity at a point is equal to the time average of the product of the perturbed

pressure and velocity. For a pressure and velocity that are random processes, this temporal

average must be taken in the limit as the characteristic timeT ! 1:

I(x; y; z) = hp(x; y; z; t)~u(x; y; z; t)i = limT!1

1

T

TZ

0

p(x; y; z; t)~u(x; y; z; t) dt (E.1)

where~u(x; y; z; t) is the acoustic velocity vector. This temporal average may be replaced by

an expected value:

I(x; y; z) = hp(x; y; z; t)~u(x; y; z; t)i

= E[p(x; y; z; t)~u(x; y; z; t)](E.2)

by assuming that the processes areergodic in correlation. In general, the relationship between

the acoustic pressure and velocity is not known and must be determined concurrently. However,

in the acoustic far-field, the radial velocity is dominant and the propagating acoustic waves

appear planar. Consequently, the relationship between the acoustic pressure and velocity can

be approximated by

~u(R; �;�; t) � ur(R; �; �; t) =p(R; �; �; t)

�c(E.3)

where the spatial description has been changed to spherical coordinates for consistency with

equations (7.18), (7.35), or (7.37). Thus, the far-field radiated acoustic intensity becomes

It(R; �; �) =1

�cE�p2t (R; �; �; t)

�=

p2t

�c(E.4)

where the mean square acoustic pressure is given by one of the equations referenced above.

The acoustic power incident upon a surface is defined as the integral of the normal acoustic

intensity over the surface:

�s =

Z

S

I(r; �; �) � n dS (E.5)

258

wheren is the surface unit normal. Therefore, the total acoustic power radiated by the plate is

calculated by integrating the acoustic intensity over a far-field hemisphere as follows.

�r =

2�Z

�=0

�=2Z

�=0

I(r; �; �) R2 sin � d� d� (E.6)

where the radiated intensity has the form given in equation (E.4). Note that theR2 in the

integrand cancels with that in the expression for the far-field acoustic intensity, equations

(7.18)/(7.35)/(7.37) and (E.4). This integral must be evaluated numerically, which can be

accomplished via Gauss-Legendre numerical integration (see Appendix B).

Recall that for the case of acoustic pressure excitation the incident acoustic pressure consists

of obliquely incident plane waves. The incident acoustic intensity is then, from the above

discussion,

Ii =p2i�c

(E.7)

and the incident acoustic power is simply the product of the normal acoustic intensity and the

area of the plate

�i =p2i ab cos �i

�c(E.8)

Finally, the transmission loss (TL) is defined in units of dB by the logarithm of the ratio of

incident acoustic power to radiated acoustic power

TL = 10 log

��i

�r

�(E.9)

259

Appendix FContinuous Spectral Examples

A continuous spectral analysis was one approach used in Chapter 6 to derive the expression

for the dynamic response correlation matrices of a panel-type structure subjected to elevated

and/or one of the following mechanical loads; acoustic pressure, concentrated forces, or base

acceleration. A similar approach was used in Chapter 7 to derive an expression for the mean

square radiated acoustic pressure due to the same structure, but limited to two of the three

mechanical loads; acoustic pressure or concentrated forces. The dynamic response correlation

matrices and mean square radiated pressure require evaluation of spectral integrals of the forms

given in equations (6.36) and (7.15), respectively. It is clear that these equations have a very

similar form and will be denoted byI(d;v;a)mn (�) in the following generic expression

I(d;v;a)mn (� ) =

1Z

�1

!(0;2;4)H�

m(!)Hn(!)S��(!)ei!� d!

=

1Z

�1

!(0;2;4)S��(!)

(!2m � !2

� i2�m!m!)(!2n � !2 + i2�n!n!)

ei!� d!

(F.1)

where the expressions for the frequency response functionsH�(!) have been substituted from

equation (6.29) andS��(!) is the power spectral density function of the excitation. The

superscript(d; v; a) indicates that the multiplier of!(0;2;4) corresponds to displacement, velocity,

and acceleration response, respectively.

Two sample cases for the evaluation of the integral in equation (F.1) are presented in this

appendix including harmonic and band-unlimited white noise excitation. The case of harmonic

excitation is obviously not stochastic, but is included as a simple comparative case.

260

F.1 Harmonic Excitation

Harmonic excitation can be described in either of the following forms

S��(!) =

� S0 j!j = !0

0 otherwise, S��(!) = S0 �(j!j � !0) (F.2)

or

C��(� ) = 2S0 cos!0� (F.3)

whereS0 is a constant,�(j!j � !0) is a spectral Dirac delta function,� = t2 � t1, and the

subscript�� indicates the excitation could be due to acoustic pressure, concentrated forces, or

base acceleration. Substitution ofS��(!) into equation (F.1) results in

I(d;v;a)mn (� ) =

1Z�1

!(0;2;4)

(!2m � !2 � i2�m!m!)(!2

n � !2 + i2�n!n!)S0�(j!j � !0)e

i!� d! (F.4)

This equation may be simplified by retaining only even terms, because the integral of odd terms

over even limits vanishes, as

I(d;v;a)mn (� ) = 2

1Z0

!(0;2;4)

��!2m � !2

��!2n � !2

�+ 4�m�n!m!n!

2�cos!�h

(!2m � !2)

2+ 4�2m!2

m!2ih(!2

n � !2)2+ 4�2n!

2n!

2i

+2!(1;3;5)

��m!m

�!2n � !2

�� n!n

�!2m � !2

��sin!�h

(!2m � !2)

2+ 4�2m!2

m!2ih(!2

n � !2)2+ 4�2n!

2n!

2i!S0�(! � !0) d!

(F.5)

Enforcing the Dirac delta function results in

I(d;v;a)mn (� ) =

2S0!(0;2;4)0

��!2m � !2

0

��!2n � !2

0

�+ 4�m�n!m!n!

20

�h�!2m � !2

0

�2+ 4�2m!2

m!20

ih�!2n � !2

0

�2+ 4�2n!

2n!

20

i cos!0�

+4S0!

(1;3;5)0

��m!m

�!2n � !2

0

�� n!n

�!2m � !2

0

��h�!2m � !2

0

�2+ 4�2m!2

m!20

ih�!2n � !2

0

�2+ 4�2n!

2n!

20

i sin!0�

(F.6)

This expression can be used directly for calculation of the dynamic response correlation matrices

given by equation (6.38). The mean square radiated pressure makes use of the acceleration

261

response case, i.e.,I(a)mn(� ), by making the substitution

� =(� 0� �)K + (�0

� �)�

c(F.7)

and using the result in equation (7.18). Finally, the mean square dynamic responses can be

obtained from the diagonal elements of the dynamic response correlation matrices for zero time

difference(� = 0) which get input from the following simplified form forI(d;v;a)mn (0)

I(d;v;a)mn (0) =

2S0!(0;2;4)0

��!2m � !2

0

��!2n � !2

0

�+ 4�m�n!m!n!

20

�h�!2m � !2

0

�2+ 4�2m!2

m!20

ih�!2n � !2

0

�2+ 4�2n!

2n!

20

i (F.8)

F.2 Band-Unlimited White Noise Excitation

Band-unlimited white noise excitation may be described by either of the two following forms:

S��(!) = S0 (F.9)

or

C��(� ) = 2�S0�(� ) (F.10)

whereS0 is a constant for all frequencies and� = t2 � t1. Substitution ofS��(!) into equation

(F.1) results in

I(d;v;a)mn (� ) =

1Z�1

!(0;2;4)S0

(!2m � !2

� i2�m!m!)(!2n � !2 + i2�n!n!)

ei!� d! (F.11)

One way to evaluate the integral in equation F.11 is to employ the Cauchy residue theorem by

integrating over a semicircular contour, in the limit as the radius of the contour becomes infinite,

in the upper half plane (for� > 0) enclosing the poles of the integrand in the UHP. The case

for I(a)mn(� ), i.e., acceleration response—!4 in the numerator, in equation (F.11) is unbounded.

Therefore, attention is limited toI(d;v)mn (� ), the cases for!(0;2) in the numerator. A schematic

of the contour of integration is shown in Figure F.1.

262

-R R

ωdn+iξn-ωdn+iξn

CR

iξ

ωd

Figure F.1 Schematic of upper-half-plane contour for evaluation ofdynamic response due to band-unlimited white noise excitation.

The integral around the closed contour can be written as

I

C

f (!)ei!�d! =

RZ

�R

f(!)ei!�d! +

Z

CR

f(!)ei!�d! (F.12)

where the integrand in equation (F.11) has been denoted byf(!)ei!� andCR denotes the circular

portion of the contour. In the limit asR ! 1, the first integral on the right-hand side takes

the desired form, the Cauchy principal value, and the sum of the two integrals can be evaluated

by employing the Cauchy residue theorem as

limR!1

RZ

�R

f(!)ei!� d! + limR!1

Z

CR

f(!)ei!� d! = 2�iNrXk=1

Res�f(!)ei!� ; rk

�(F.13)

where the expression on the right of the summation represents the residue off(!)ei!� at the

pole rk andNr is the number of poles off(!) enclosed within the contour (which becomes all

of those in the UHP asR !1). Recall the expression forf(!) shown in equation (F.11). This

263

expression can be rewritten in terms of simple poles as follows

f(!) =!(0;2)S0

(!2m � !2

� i2�m!m!)(!2n � !2 + i2�n!n!)

=!(0;2)S0

(! � !1)(! � !2)(! � !3)(! � !4)

(F.14)

where

!1;2 = �i�m!m �q��2m!

2m + !2

m and !3;4 = i�n!n �q��2n!

2n + !2

n

= �!m

q1 � �2m � i�m!m = �!n

q1 � �2n + i�n!n

= �!dm � i�m = �!dn + i�n

(F.15)

Becausef(!) can be written as

f(!) ='i(!)

(! � !i)for i = 1 ! 4 (F.16)

the residue off(!) at the pole!i is given by the coefficient of1=(! � !i) in the Laurent

expansion (see Churchill and Brown [61]) given by

f(!) ='i(!)

(! � !i)+

'0

i(!)

1!+ � � � (F.17)

Thus, the integral on the closed contour becomes

limR!1

RZ�R

f(!)ei!�d! + limR!1

ZCR

f(!)ei!�d! = 2�i�lim!!!3

�(! � !3)f(!)e

i!�

+ lim!!!4

�(! � !4)f(!)e

i!�� (F.18)

It can be shown that the integral on the circular portion of the contour is zero (see Hildebrand

[62]) so the desired result has been achieved as

I(d;v)mn (� ) =

1Z�1

f(!)ei!� d!

= 2�i�lim!!!3

�(! � !3)f(!)e

i!�+ lim

!!!4

�(! � !4)f(!)e

i!��

= 2�i

"!(0;2)3 S0e

i!3�

(!3 � !1)(!3 � !2)(!3 � !4)+

!(0;2)4 S0e

i!4�

(!4 � !1)(!4 � !2)(!4 � !3)

# (F.19)

264

The terms in this expression can the expanded into a sum of real and imaginary components

and simplified to the following

I(d)mn(� ) =

4�(�m + �n)S0e��n� cos!dn�h

(!dm � !dn)2 + (�m + �n)

2ih(!dm + !dn)

2 + (�m + �n)2i

�

4�h!2dn � !2

dm � (�m + �n)2iS0e

��n� sin!dn�h(!dm � !dn)

2 + (�m + �n)2ih(!dm + !dn)

2 + (�m + �n)2i

(F.20)

and

I(v)mn(� ) =

4��!2dn�m + !2

dm�n + �n�2m + �2n�m

�S0e

��n� cos!dn�h(!dm � !dn)

2(�m + �n)2ih(!dm + !dn)

2(�m + �n)2i

�

4�h!4dn�!2

dn!2dm�!2

dn�2m+2!2

dn�m�n+!2dm�

2n+�2n(�m+�n)

2iS0e

��n� sin!dn�h(!dm � !dn)

2(�m + �n)2ih(!dm + !dn)

2(�m + �n)2i

(F.21)

These expressions can be used directly for calculation of the dynamic response correlation

matrices given by equation (6.38). The mean square radiated pressure cannot be computed in

this case as the acceleration response for band-unlimited white noise is unbounded, as stated

previously. Finally, the mean square dynamic responses can be obtained from the diagonal

elements of the dynamic response correlation matrices for zero time difference(� = 0) which

get input from the following simplified forms forI(d)mn(0) and I(v)mn(0)

I(d)mn(0) =

4�S0(�m + �n)h(!dn � !dm)

2+ (�m + �n)

2ih(!dn + !dm)

2+ (�m + �n)

2i (F.22)

and

I(v)mn(0) =

4�S0

��m

�!2dn + �2n

�+ �n

�!2dm + �2m

��h(!dn � !dm)2 + (�m + �n)

2ih(!dn + !dm)2 + (�m + �n)

2i (F.23)

265

Vita

Travis L. Turner was born in Hampton, Virginia on February 26, 1965. He was raised in York

County, Virginia, where he graduated from Tabb High School in June of 1983. He then entered

the School of Engineering at Old Dominion University where he received a Bachelor of Science

degree in Mechanical Engineering in 1987. He was hired by NASA Langley Research Center

(LaRC) immediately following his undergraduate studies. He pursued graduate studies while

employed at NASA LaRC and received a Master of Science degree in Engineering Mechanics

from Old Dominion Univeristy in 1991. He then completed an academic residency at Virginia

Polytechnic Institute and State University in 1992 under the support of NASA LaRC. Since

returning to NASA he has worked in the areas of geometrically nonlinear vibration, thermal

buckling and post-buckling, and adaptive/smart structures involving SMAs and piezoelectrics.

266

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